I feel that several different issues are being rolled together here which is confusing.
DC, half wave rectification etc visibly distort the sine wave that is household current, you can see that on a 'scope. I can see how a well constructed amplifier power supply will isolate the audio side from these variations.
However one issue that remains unresolved in my mind is that of high or radio frequency interference entering the amplifier through the mains. RF in particular appears to be able to get into the active circuits of an amplifier despite the normal isolating effects of the power supply.
I have detailed, in another thread, my experience of using a filtered power supply to feed a high quality but modestly powered system. Conditions were extreme, hi-fi show with a hell of a lot of equipment on the same supply. The power supply removed a level of 'grain' or 'haze' that could be heard in the system in that specific situation.
Entirely subjective evaluation of course, but several people heard the improvement without knowing what it was, including my non enthusiast boss who was happy to buy the filter after the show. Not conclusive by any means, but this was the one and only time ever, that I was convinced that any mains product produced a definite benefit.