Is there a way to always view trade route paths?

Solution 1:

Yes, you can see the trade routes by enabling the Trade Routes option in the map options, the same place you enable Resource Icons, Yield Icons, and the Hex Grid. Screenshot below shows a trade route leading north-northwest out of Salvador. Because I'm playing Brazil the line is green and it's hard to see it on the map, but you can see it best underneath the red Austrian scout unit at left. The little arrows are animated. The routes are also visible on the strategic overview map.

In addition you can also see trade routes by hovering the mouse over a trade unit. I've only seen this when looking at another civilization's ship, I'm not clear if you can get this type of display for your own trade route units. This one looks to be the same graphical effect as the video you posted.