Drum brake installation troubles...

...Switching shoes side to side is not going to bring the material further up on the steel.

Actually, it will. Because if you swap shoes from the correct side to the incorrect side the front shoe will be on the back of the axle, and the back shoe will be on the front of the axle. Or, if he has only completed one side, it's possible to put both back or both front shoes on one side. Not saying that this is the OP's problem for sure, just something to check.