In Round Earth Theory, during the Total Solar Eclipse the Moon is in alignment with the Sun and Earth, on the Ecliptic, so I would expect the shadow to appear on the line of the Ecliptic upon the Earth -- the plane of the Sun-Earth system.
Right. I see what you're saying. This is all horribly out of scale but I think it illustrates the principle of how eclipses can appear at different latitudes:
In the top diagram the sun, moon and earth are
exactly aligned. In that case yes, the shadow appears on that plane.
In the second diagram I've moved the moon up a few pixels - that represents a tiny misalignment but small enough that the shadow still hits the earth, just at a different latitude. In that example it's hitting the north pole and you'll note that the shadow covers a far larger area - the width of the shadow cast by the moon in a vertical direction isn't different, it's just because of the angle of the ground with respect to that shadow it coverers a wider area.
In the third diagram I've moved the moon up further and that's when the shadow misses the earth completely, which is what happens most of the time.
Then you have the complication of the earth spinning. The speed of the ground varies with latitude of course whereas the shadow moves at a constant speed, so that complicates how the path moves. Again, you're mapping a 2D disc of the shadow path on to a 3D spinning object. There is some complexity here.
Your attitude seems to be if you don't understand something then it can't be true. You've said that eclipse paths make more sense on a FE, but all you have to back that up is a series of arcs drawn on a FE map, but you've agreed elsewhere that there is no definitive FE map. I mentioned the Santiago to Sydney flight which was in the air as I was typing, the route of which makes no sense on that map. I note those images come from the Wiki, I had a look at the page about it and it gives no details about how eclipses work on a FE - there's no diagram which explains it like there is for RE.