This is my understanding of the subject.
Live music are complex vibration waves in air, an event which occur in space and time. Real live music can only happen as "pure class A" events in a mechanical sense. The pioneering audio amplifier engineers got it all right in the beginning. Back then there were mainly SE pure class A tube amps of very low power driving horn load or similar designs high efficiency speakers. (The success/shortcomings of such early speakers is another story for another A'gon thread.)
IMHO the reproduction of this real class A mechanical music event(recorded) can't get closer than through a pure class A electronic amplifier. I will not attempt to enter into the argument of "distortions" harmonic or others, of different types of amp designs, because honestly I cannot hear them. This is due, I can fairly suspect to the real event recording is not distortion free either, nor is my listening room perfect. So the minute amp distortions, added to the overwhelming inherent distortions, has become a non issue. In my own experience, my current SE pure Class A monoblocs provide that uncanny "breath" and "feel" in space and time which blew away all my previous equipment of various classes, AB & B etc. Ironically they all have more impressive distortion measurements in the catalogs than the Class A. Apparently the distortion numbers had not affect the way the "breath" and "feel" being reproduced. Don't take my word fully, go try out some pure Class A amp in your home setup to experience it. Sorry, you may need another set of speakers too.
I earn my bread as an electrical techy, but for my audiophile pursuits I will push aside my scopes and meters and use my ears. I wonder whether the audiophile equipment tradition had it all wrong, paying to much attention to measuring the unimportant parameter. I have no problem with the various class modes of amp operation, but today i take a break cos the posts above done a wonder job. I will attempt it in lay terms.
IMHO classes AB,B,C etc will always be commercial or technical compromises. Some came into scene to drive low efficiency modern speakers. Some such designs rightly belong in professional audio or even public announcing systems, not home audio. Chopping pure signal apart, reflecting mirror images of it, massaging it with some feedback at some point, and then reconstitute it for consumption. This is not real milk.
IMO pure Class A, warts and all is the way to go for the purist audiophile.