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.
OK. But why is North-South movement an issue? The rotation of the earth is at an angle so while the shadow may go in a straight line in the East-West direction, that angle means the shadow isn't going to hit the earth at the same latitude as it does so. Plus as I said the earth is rotating as it does so, the ground speed varies by latitude so that adds some complication.
And as I said you get very wide shadows when eclipses appear near the poles - it's not the size of the shadow that's different (although that does change a bit as the moon's distance from us varies - hence annular eclipses when it's too far away to completely obscure the sun), it's the angle of the ground with respect to that shadow that changes.
There is some complication here, admittedly, but a lot of your issues with RE seem to come from you not understanding them and thus concluding they must be impossible.