Is it not possible with both Class d and Class A/B amps to engineer the amp to the lowest possible distortion and then add back enough 2nd and 3rd harmonics to give the amp the degree of musicality desired? And if so, could a "distortion" dial be added to allow the user to fine tune the sound to their liking?
Theoretically, to the former, possibly ’yes’. To the latter, likely ’no’ since such a control would have to modify some parameter of the amplifier that might be critical and not take kindly to such an adjustment.
I wouldn’t hold your breathe for such a thing. Usually designers of solid state are very conscious of the ’numbers guys’ that only look at specs and poo-poo the listening experience, even if the latter is what its all about. So they often try to get the distortion as low as they can and anything that raises it is anathema. There are a few that are careful about how their designs make distortion; Nelson Pass is an example.