Please help me understand

The sound is in the ear of the beholder.

Danoroo,

Music is just much more complex than any test while warm or bright is just different types of distortion. Why do you think amps should sound the same? What is to my liking is not necessarily your cup of tea. I like neutral sound while you may prefer warm sound. There is also synergy with the speakers and the rest of the system including listening room. In addition, since there is no perfect amp, some amps are better than others for given type of music. It would be nice to know ahead, performing some type of complex test, but as it stands now even known types of distortions (like TIM responsible for brightness) are not listed, being often too cumbersome to measure.

"some of them aren't doing their job very well"

Yes some of them are poor, but often you face compromise for the type of music you like - unless you have a lot of money (there are amps up to $1,000,000) and it can still have poor synergy with your system. Trust your ears and quit reading technical specifications because often amps with the best specs have the worst sound (and there are technical reasons for that).

That's just one concept, actually.

Dynamics.slew rates. current vs.voltage are some things that come to mind that will affect sound and not distortion figures.