Can't set stroke animation start point on a rectangle

Yes, because radius is uniform the cross-section of corners is exactly like cross-section of straight segments - so it looks badly deformed.
But this is a separate problem entirely. It’s not something built-in into Blender, but there are plenty of discussions and solutions around.

This user for example has something called “Even Curve to Mesh” in their nodes (I don’t know if it’s still up to date):