Circle loading animation

Here's my SVG-only version. The background colour wheel isn't perfect, but I think I got fairly close.

<svg width="135" height="135" viewBox="0 0 200 200">
    <defs>
        <filter id="blur" color-interpolation-filters="linear">
            <feGaussianBlur in="SourceGraphic" stdDeviation="11"/>
        </filter>
        <mask id="mask">
            <circle cx="0" cy="0" r="90" fill="white"/>
        </mask>
        <linearGradient id="gloss" x2="0" y2="0.4">
            <stop offset="0" stop-color="white" stop-opacity="0.5"/>
            <stop offset="1" stop-color="white" stop-opacity="0"/>
        </linearGradient>
    </defs>

    <g transform="translate(100,100)" mask="url(#mask)">
        <g filter="url(#blur)">
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#c44"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#c09" transform="rotate(30)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#c0c" transform="rotate(60)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#90c" transform="rotate(90)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#44c" transform="rotate(120)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#09c" transform="rotate(150)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#0cc" transform="rotate(180)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#0c9" transform="rotate(210)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#4c4" transform="rotate(240)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#9c0" transform="rotate(270)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#cc0" transform="rotate(300)"/>
            <polygon points="0,0, -100,-26.8, -100,26.8" fill="#c90" transform="rotate(330)"/>
        </g>
        <g transform="scale(0.9,0.9)">
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4"/>
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4" transform="rotate(60)"/>
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4" transform="rotate(120)"/>
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4" transform="rotate(180)"/>
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4" transform="rotate(240)"/>
            <path d="M0,0C5,-61,-32,-86,-95,-90L-100,-46C-65,-53,-24,-35,0,0Z" fill="black" fill-opacity="0.4" transform="rotate(300)"/>
            <animateTransform attributeType="xml" attributeName="transform" type="rotate" from="0" to="360" dur="4s" repeatCount="indefinite"/>

        </g>
        <circle r="83" fill="url(#gloss)"/>
        <circle r="90" fill="none" stroke="black" stroke-width="2"/>
    </g>
</svg>

Here's my effort. The conical gradient is an embedded bitmap image extracted by calculating the maximum value of each pixel in the animated GIF posted by the OP. A semi-opaque black windmill pattern is superimposed on top of that and animated, and a blur filter gets rid of the JPEG artefacts.

(Edit: Added a reflective highlight to make it look a bit more 3D)

<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"
  width="121" height="121" viewBox="0 0 121 121">
  <defs>
    <clipPath id="circ">
      <circle r="60" cx="60.5" cy="60.5"/>
    </clipPath>
    <linearGradient id="shine" x1="0%" y1="0%" x2="0%" y2="100%">
      <stop offset="0%" style="stop-color:#fff;stop-opacity:0.6" />
      <stop offset="10%" style="stop-color:#fff;stop-opacity:0.3" />
      <stop offset="20%" style="stop-color:#fff;stop-opacity:0.1" />
      <stop offset="40%" style="stop-color:#fff;stop-opacity:0" />
    </linearGradient>
    <filter id="blur">
      <feGaussianBlur in="SourceGraphic" stdDeviation="2"/>
    </filter>
  </defs>
  <image width="121" height="121" filter="url(#blur)" xlink:href="data:image/jpeg;base64,
  /9j/4AAQSkZJRgABAQEASABIAAD/2wBDACAWGBwYFCAcGhwkIiAmMFA0MCwsMGJGSjpQdGZ6eHJm
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  AhEBAxEB/8QAGQAAAwEBAQAAAAAAAAAAAAAAAAECBAMF/8QAGAEAAwEBAAAAAAAAAAAAAAAAAAEC
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  AAwDAQACAAMAAAAQCZ9GgAVrOID+MKOo0R0OMKN+ec1rZcb1FaGkEgox777+BDUwvvfOejgbCEVt
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  EOHlsItTT9kstHnGIiODw4/eERZEMU6PAs4PrpyO+S3mz1vlttvdiWwXtvNtlzINvNlllDDDt5my
  ywyhvmGOyy2wwwwYcBZbYYYa7u/ktsMfjCGMz7w98s/0D9/J/8QAIxABAAICAgICAgMAAAAAAAAA
  AQARECExQSBRMGGhsXGBkf/aAAgBAQABPxDCgWxl/q4naJ0QW2rDhCfVnIGK21QfaF0xz/bgiWeL
  pKCIXTuzfbRYZLyX7mygIbujDSWOVotlzdeX3KUc/EAKA8y1W3DBssxuGW2eX4AL4dpnCbP68Khw
  NHixgcD4sUjkpn3zhfcvwBJDDgcV4K6rkl/UVS5cuXLhDpgtwMuEduFeFSmWl1yKOZUqVKm5WKlS
  sCCbXiB6Sh4jLgrwrJULQ6E6IFFZVisKw/mjdRdmQgjsglQalmFiokfqrtn1zQkAvwAwiomBwGGk
  1BgSymPWb/iBAhgBKlYGGEiCWv5gUUZJgsY7e36ylagQIEqJEggqXoV+8IgoPFBKYS229RtpD7jH
  hIYYg5SFtM/cMbbeoFFHwc04MnGHKcXl/9k=" clip-path="url(#circ)" />
  <g transform="translate(60.5,60.5)">
    <path d="M0 0A56 56 0 0 0 0 56 56 56 0 0 0 32.916 45.305 56 56 0 0 1 0 0 56
        56 0 0 0 53.259 17.305 56 56 0 0 0 53.259-17.305 56 56 0 0 1 0 0 56 56
        0 0 0 32.916-45.305 56 56 0 0 0 0-56 56 56 0 0 1 0 0 56 56 0 0 0
        -32.916-45.305 56 56 0 0 0-53.259-17.305 56 56 0 0 1 0 0 56 56 0 0 0
        -53.259 17.305 56 56 0 0 0-32.916 45.305 56 56 0 0 1 0 0Z"
        stroke="none" fill="#000" opacity="0.25" transform="rotate(0)">
      <animateTransform attributeName="transform" type="rotate" from="0"
          to="72" begin="0s" dur="0.6s" repeatCount="indefinite" />
    </path>
    <circle r="59" stroke="#000" stroke-width="2" fill="none" opacity="0.25" />
    <circle r="55" fill="url(#shine)" stroke="none" />
  </g>
</svg>

Canvas Approach

Since this is a loading animation where the dimensions are probably going to be fixed with almost nil user interactions, Canvas would also be a good option because it doesn't add any extra elements to the DOM. Canvas drawing commands are pretty similar to SVG and the browser support is also not worse.

One disadvantage would be that Canvas does not have its own blur filter (unlike SVG). But that can be overcome by either using CSS blur filter (has very low browser support) or the libraries mentioned in this Stack Overflow thread.

Background Gradient Wheel:

The background gradient wheel is created using an approach similar to the one detailed in my answer here. Basically we find multiple points in the circle and draw lines each of which have different colored stroke. By modifying the hue value for every line we can paint the gradient wheel.

In the below screenshot the first picture shows how the background would have looked if we had drawn just 24 lines (with a hue change of 15 between every line)and the second one is our actual gradient wheel which has 360 lines in total with the hue being incremented by 1 for each line.

Fan:

The fan is created using the same approach as used in your SVG snippet. Path commands are used to draw each spoke. While use tag is used in SVG to repeat a shape, loops can be used in Canvas.

The main difference here between SVG and Canvas is that Canvas cannot take in transform origin as a parameter for the rotate function and so context must be translated to the center point before applying rotations.

Finally the canvas must be clipped into a circle because the default shape is a square (as height and width are same). The below screenshot shows the unclipped and clipped versions of the fan.

This fan is then placed on top of the background gradient wheel.

3D effect:

The 3D effect on top is provided by adding a small arc with an higher amount of transparency over the background and the fan.

Below is the screenshot of the full picture without any animation.

Animation:

Animation is added by using the window.requestAnimationFrame method which calls the function passed as an argument at regular intervals. This method would generally call the function about 60 times per second (according to MDN). By incrementing the value of the counter variable during every iteration and adding it to the angle of the fan's spokes the animation effect can be achieved.

window.onload = function() {
  var canvas = document.querySelector("#canvas"),
    ctx = canvas.getContext("2d"),
    counter = 360;

  function fan() {
    ctx.clearRect(0, 0, 100, 100);
    for (var i = 0; i < 360; i++) {
      ctx.strokeStyle = "hsl(" + (180 - i) + ", 60%, 50%)";
      ctx.beginPath();
      ctx.moveTo(50, 50);
      x = 50 + 50 * Math.cos((i / 360) * Math.PI * 2);
      y = 50 + 50 * Math.sin((i / 360) * Math.PI * 2)
      ctx.lineTo(x, y);
      ctx.lineWidth = 2;
      ctx.stroke();
    }
    counter++;
    for (var j = 0; j < 6; j++) {
      ctx.save();
      ctx.beginPath();
      ctx.arc(50, 50, 50, 0, Math.PI * 2, true);
      ctx.clip();
      ctx.translate(50, 50);
      ctx.rotate(((60 * j) + counter) * Math.PI / 180);
      ctx.beginPath();
      ctx.moveTo(0, 0);
      ctx.bezierCurveTo(0, 0, 30, 50, 100, 0);
      x = 75 * Math.cos((-20 / 360) * Math.PI * 2);
      y = 75 * Math.sin((-20 / 360) * Math.PI * 2)
      ctx.lineTo(x, y);
      ctx.bezierCurveTo(x, y, (x - 30), (y + 40), 0, 0);
      ctx.closePath();
      ctx.fillStyle = "rgba(0,0,0,0.5)";
      ctx.fill();
      ctx.restore();
    }
    ctx.save();
    ctx.beginPath();
    ctx.arc(50, 50, 50, 0, Math.PI, true);
    ctx.arc(50, 55, 50, Math.PI, 0, false);
    ctx.fillStyle = "rgba(0,0,0,0.15)";
    ctx.closePath();
    ctx.fill();
    ctx.restore();
    window.requestAnimationFrame(fan);
  }
  fan();
}

<canvas width='100px' height='100px' id='canvas'></canvas>

SVG Approach

The same approach as described above can be used with SVG also. The only downside would be the no. of extra elements that get added to the DOM both for the background and the fan.

window.onload = function() {
  var colorWheel = document.querySelector("#color-wheel");
  for (var i = 0; i < 360; i++) {
    lineColor = "hsl(" + (180 - i) + ", 60%, 50%)";
    x = 50 + 50 * Math.cos((i / 360) * Math.PI * 2);
    y = 50 + 50 * Math.sin((i / 360) * Math.PI * 2);
    line = document.createElementNS("http://www.w3.org/2000/svg", "line");
    line.setAttribute('x1', 50);
    line.setAttribute('y1', 50);
    line.setAttribute('x2', x);
    line.setAttribute('y2', y);
    line.setAttribute('stroke', lineColor);
    line.setAttribute('stroke-width', 2);
    colorWheel.appendChild(line);
  }
}

<svg class="rotating-spinners" width="100px" height="100px" viewBox="0 0 100 100" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
  <defs>
    <path id="spin-part" class="spike" d="M 50,50 c0,0 30,50 100,0 l-5,-35 c0,0 -30,50 -95,35Z" />
    <clipPath id="shape">
      <circle r="50" cx="50" cy="50" />
    </clipPath>
    <clipPath id="shade">
      <path d='M-5,55 a55,55 1 0,1 110,0 h-5 a50,50 1 0,0 -100,0' />
    </clipPath>
  </defs>
  <g id='color-wheel' clip-path='url(#shape)'>
  </g>
  <g id='fan' fill-opacity="0.5" clip-path='url(#shape)'>
    <use x="0" y="0" xlink:href="#spin-part" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(60, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(120, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(180, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(240, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(300, 50, 50)" />
    <animateTransform attributeType="xml" attributeName="transform" type="rotate" from="0,50,50" to="360,50,50" dur="6s" repeatCount="indefinite" />
  </g>
  <circle r='50' cx='50' cy='50' fill-opacity='0.15' clip-path='url(#shade)' />
</svg>

Mixed Approach

Or, if you have no problems with the extra elements for the fan but just want to avoid the 360 line elements that would get added, you could use a mixture of Canvas (for the background) and SVG for the fans like in the below snippet.

window.onload = function() {
  var canvas = document.querySelector("#canvas");
  var ctx = canvas.getContext("2d");

  for (var i = 0; i < 360; i++) {
    ctx.strokeStyle = "hsl(" + (180 - i) + ", 60%, 50%)";
    ctx.beginPath();
    ctx.moveTo(50, 50);
    x = 50 + 50 * Math.cos((i / 360) * Math.PI * 2);
    y = 50 + 50 * Math.sin((i / 360) * Math.PI * 2)
    ctx.lineTo(x, y);
    ctx.lineWidth = 2;
    ctx.stroke();
  }
  ctx.save();
  ctx.beginPath();
  ctx.arc(50, 50, 50, 0, Math.PI, true);
  ctx.arc(50, 55, 50, Math.PI, 0, false);
  ctx.fillStyle = "rgba(0,0,0,0.15)";
  ctx.closePath();
  ctx.fill();
  ctx.restore();
}

div {
  position: relative;
  height: 100px;
  width: 100px;
}
canvas,
svg {
  position: absolute;
  top: 0px;
  left: 0px;
  height: 100%;
  width: 100%;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/prefixfree/1.0.7/prefixfree.min.js"></script>
<div class='container'>
  <canvas width='100px' height='100px' id='canvas'></canvas>
  <svg class="rotating-spinners" width="100%" height="100%" viewBox="0 0 100 100" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
    <defs>
      <path id="spin-part" class="spike" d="M 50,50 c0,0 30,50 100,0 l-5,-35 c0,0 -30,50 -95,35Z" />
      <clipPath id="shape">
        <circle r="50" cx="50" cy="50" />
      </clipPath>
    </defs>
    <g id='fan' fill-opacity="0.5" clip-path="url(#shape)">
      <use x="0" y="0" xlink:href="#spin-part" />
      <use x="0" y="0" xlink:href="#spin-part" transform="rotate(60, 50, 50)" />
      <use x="0" y="0" xlink:href="#spin-part" transform="rotate(120, 50, 50)" />
      <use x="0" y="0" xlink:href="#spin-part" transform="rotate(180, 50, 50)" />
      <use x="0" y="0" xlink:href="#spin-part" transform="rotate(240, 50, 50)" />
      <use x="0" y="0" xlink:href="#spin-part" transform="rotate(300, 50, 50)" />
      <animateTransform attributeType="xml" attributeName="transform" type="rotate" from="0,50,50" to="360,50,50" dur="6s" repeatCount="indefinite" />
    </g>
  </svg>
</div>

I had to do this with a mixture of SVG and CSS gradients which i know is against the request but is what i know. I used some of your original code, mostly the SVG parts for the propeller shapes.

The radial gradient is made using 12 li elements.

.wheel,
.umbrella,
.color {
  content: "";
  position: absolute;
  border-radius: 50%;
  width: 15em;
  height: 15em;
  margin: 0;
  padding: 0;
}
.wheel {
  overflow: hidden;
  width: 15em;
  height: 15em;
  position: relative;
}
.umbrella {
  position: relative;
  -webkit-transform: scale(1.35);
}
.color,
.color:nth-child(n+7):after {
  clip: rect(0, 15em, 15em, 7.5em);
}
.color:after,
.color:nth-child(n+7) {
  content: "";
  position: absolute;
  border-radius: 50%;
  left: calc(50% - 7.5em);
  top: calc(50% - 7.5em);
  width: 15em;
  height: 15em;
  clip: rect(0, 7.5em, 15em, 0);
}
.color:nth-child(1):after {
  background-color: #9ED110;
  transform: rotate(30deg);
  z-index: 12;
}
.color:nth-child(2):after {
  background-color: #50B517;
  transform: rotate(60deg);
  z-index: 11;
}
.color:nth-child(3):after {
  background-color: #179067;
  transform: rotate(90deg);
  z-index: 10;
}
.color:nth-child(4):after {
  background-color: #476EAF;
  transform: rotate(120deg);
  z-index: 9;
}
.color:nth-child(5):after {
  background-color: #9f49ac;
  transform: rotate(150deg);
  z-index: 8;
}
.color:nth-child(6):after {
  background-color: #CC42A2;
  transform: rotate(180deg);
  z-index: 7;
}
.color:nth-child(7):after {
  background-color: #FF3BA7;
  transform: rotate(180deg);
}
.color:nth-child(8):after {
  background-color: #FF5800;
  transform: rotate(210deg);
}
.color:nth-child(9):after {
  background-color: #FF8100;
  transform: rotate(240deg);
}
.color:nth-child(10):after {
  background-color: #FEAC00;
  transform: rotate(270deg);
}
.color:nth-child(11):after {
  background-color: #FFCC00;
  transform: rotate(300deg);
}
.color:nth-child(12):after {
  background-color: #EDE604;
  transform: rotate(330deg);
}

<div class="wheel">
  <ul class="umbrella">
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
  </ul>
</div>

These 12 elements can then be blurred together to form the smooth gradient.

I then animated the spin parts to make the effect you require.

var rotation = 0;

$(document).ready(function() {
  setInterval(function() {
    rotation += 1;
    $('.wheel svg').css({
      'transform': 'rotate(' + rotation + 'deg)'
    });;
  }, 10);
});

.wheel,
.umbrella,
.color {
  content: "";
  position: absolute;
  border-radius: 50%;
  width: 15em;
  height: 15em;
  margin: 0;
  padding: 0;
}
.wheel {
  overflow: hidden;
  width: 15em;
  height: 15em;
  position: relative;
}
.umbrella {
  position: relative;
  filter: blur(.75em);
  -webkit-filter: blur(.75em);
  -moz-filter: blur(.75em);
  -o-filter: blur(.75em);
  -ms-filter: blur(.75em);
  filter: url(#blur);
  filter: progid: DXImageTransform.Microsoft.Blur(PixelRadius='.75');
  -webkit-transform: scale(1.35);
}
.color,
.color:nth-child(n+7):after {
  clip: rect(0, 15em, 15em, 7.5em);
}
.color:after,
.color:nth-child(n+7) {
  content: "";
  position: absolute;
  border-radius: 50%;
  left: calc(50% - 7.5em);
  top: calc(50% - 7.5em);
  width: 15em;
  height: 15em;
  clip: rect(0, 7.5em, 15em, 0);
}
.color:nth-child(1):after {
  background-color: #9ED110;
  transform: rotate(30deg);
  z-index: 12;
}
.color:nth-child(2):after {
  background-color: #50B517;
  transform: rotate(60deg);
  z-index: 11;
}
.color:nth-child(3):after {
  background-color: #179067;
  transform: rotate(90deg);
  z-index: 10;
}
.color:nth-child(4):after {
  background-color: #476EAF;
  transform: rotate(120deg);
  z-index: 9;
}
.color:nth-child(5):after {
  background-color: #9f49ac;
  transform: rotate(150deg);
  z-index: 8;
}
.color:nth-child(6):after {
  background-color: #CC42A2;
  transform: rotate(180deg);
  z-index: 7;
}
.color:nth-child(7):after {
  background-color: #FF3BA7;
  transform: rotate(180deg);
}
.color:nth-child(8):after {
  background-color: #FF5800;
  transform: rotate(210deg);
}
.color:nth-child(9):after {
  background-color: #FF8100;
  transform: rotate(240deg);
}
.color:nth-child(10):after {
  background-color: #FEAC00;
  transform: rotate(270deg);
}
.color:nth-child(11):after {
  background-color: #FFCC00;
  transform: rotate(300deg);
}
.color:nth-child(12):after {
  background-color: #EDE604;
  transform: rotate(330deg);
}
body {
  padding: 5px;
}
.wheel svg {
  position: absolute;
  top: 0;
  opacity: .5;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<figure class="animation-wrapper wheel">
  <ul class="umbrella">
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
    <li class="color"></li>
  </ul>
  <svg class="rotating-spinners" width="100%" height="100%" viewBox="0 0 100 100" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
    <defs>
      <path id="spin-part" class="spike" d="M 65,-40 C 65,-40 80,20 50,50 60,40 50,-40 50,-40Z" />
    </defs>
    <use x="0" y="0" xlink:href="#spin-part" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(60, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(120, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(180, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(240, 50, 50)" />
    <use x="0" y="0" xlink:href="#spin-part" transform="rotate(300, 50, 50)" />
  </svg>
</figure>

Conical CSS Gradient

Circle loading animation

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