if it weren't for efficiency & power, would there be any reason to own a Class D amp?
Class D amps rock the category of watts per dollar, which makes them a good choice for powered subwoofers where the goal is control and damping of large drivers while imaging and spatial information doesn't matter.
Do they beat Class A in any other categories that count for sound quality?
Power and control, both of which are harder to achieve economically with Class A. Otherwise, Class D does not beat Class A in my experience comparing Class A Lamm M1.2s and Clayton M300s with Class D Ncore NC1200 via Acoustic Imagery. I owned all of those amps and compared them directly in my system over an extended period. The Class A amplifiers were way better at portraying music naturally and believably, which seemed to relate to the spatial representation of the individual sounds and instruments. The NC1200 amps also did not beat better Class AB amps, IME. Better sound quality seems to be the current goal of Class D designers, with pockets of manufacturers and owners claiming that certain topologies and products are getting closer or achieving the goal but so far, the marketplace has not seemed to fully agree in the long term.