Why does a golden angle based spiral produce evenly distributed points?

The problem with Vogel's model is that it obscures the structure of curves from which the points are derived. If you plotted it as a line instead of points it would look like a spirolateral with lines crisscrossing all over the place. In fact, as revealed by your modified drawings, it's really four evenly spaces Fermat spirals. The Fermat spiral is given by

$$ r(\theta)=\sqrt{\theta}\\ \theta\in[0,n\pi] $$

where $n$ is the number of half-turns.

To complete the the pattern, you need to concatenate the above with its anti-symmetric twin and then repeat for the number of arms desired by rotating the (dual) Fermat spiral by $\pi/\text{arms}$ for however many arms there are. Alternatively, you could take the (half) Fermat spiral and rotate it by $2\times\text{arms}$.

When programming the above you have several parameters at your disposal. Here is a pseudocode for how I do this in the complex plane

theta=0,n*pi,npts
r=sqrt(theta)
z=r*exp(i*rho*theta)
z=[-z;z]  % complete the spiral
for k=2:arms
    z=[z z*exp(i*k*pi/arms)]
end
plot(z) % using lines or dots

So the parameters are (1) the number of half turns, n, (2) the number of points on the curve, npts, (3) a stretching or density factor, rho, and (4) the number of arms. Using 2 (half) turns, 4 arms, 41 points, and a density of 1.4 I close to Vogel's pattern, but I haven't tried to fine tune it. I have not tried to incorporate $\varphi$ into the selection of the parameters.