Can anyone suggest any reasons why you shouldn't use GPS or DME to
lead your turns when approaching a fix, such as an IAF, on a non-GPS
approach? (Meaning that you don't wait for complete reversal of the
nav instrument.)

The AIM has a couple of requirements for a "complete reversal" of the
TO/FROM indicator, but they aren't instructions about how to fly an
instrument approach. Regardless, those injunctions may be outdated in
a GPS world.