It mentions it just after 3 minutes in.
No, it doesn't explain it. This is what it says at the 3 minute mark, that straight lines will look like curves at another angle:
Angle 1:
Angle 2:
This does not answer why the lines move North-South. It answers a different question of why they might appear curved.
To understand this just put Google Earth in Globe mode, define some straight lines across the globe and then go back in to map projection mode. Voila:
Basically, it's a 2D line - the shadow path - mapped on to a 3D object - the globe - which is also spinning. And the rate of that spin varies with latitude.
It's a bit complicated, but if you look at the example at just after 3 minutes in to that video you can have a go yourself to help you understand it. A lot of your FE belief seems to stem from you not understanding the RE model and thus declaring it impossible.
That's not an explanation for why it travels North-South. This is an explanation for why a line might appear straight on a globe and make those curves on a map projection. You drew the North-South line there, not explained how that could occur in RE with a Moon which moves East to West.
Also, the below are not map projections. The lines do not appear straight or consistently curved here:
http://eclipse-maps.com/Eclipse-Maps/Welcome.html