THD, SNR, frequency response and many other metrics are easy: they define either accuracy or precision. If you want more than that, add external affects hardware or DSP. The purpose of the amplifier is to amplify.
You seem to be highly inexperienced in the subject, THD, IMD etc are parameters that are measured on highly artificial signals - singke sine waves or mix of small numbers of sine waves. Real music signals more resemble white noise than clean sine waves, and there might be hundreds of different very subtle modes of distortion, that are very difficult to measure but which have very significant influence on the precepted quality of the sound.
An industry which is known to sell an enormous amount of snake oil...