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 discusses this in the section on En Route Procedures, but doesn't say
anything about instrument approaches:

5-3-5. Airway or Route Course Changes

a. Pilots of aircraft are required to adhere to airways or routes being
flown. Special attention must be given to this requirement during course
changes. Each course change consists of variables that make the technique
applicable in each case a matter only the pilot can resolve. Some variables
which must be considered are turn radius, wind effect, airspeed, degree of
turn, and cockpit instrumentation. An early turn, as illustrated below, is one
method of adhering to airways or routes. The use of any available cockpit
instrumentation, such as Distance Measuring Equipment, may be used by the
pilot to lead the turn when making course changes. This is consistent with the
intent of 14 CFR Section 91.181, which requires pilots to operate along the
centerline of an airway and along the direct course between navigational aids
or fixes.