Someone, please correct me:
I don't believe that the idea is to "want" 2nd and 3rd harmonic distortion, but rather our brains use the 2nd and 3rd harmonics that are produced by the musical instruments to perceive certain aspects from the sound. I don't believe any type of harmonic distortion from an amplifier is "desirable".
I read Atmasphere’s words as, ‘given the choice between no 2nd/3rd harmonics, then having them allows one to mask the higher order harmonics. So you want them higher than everything at 4th and higher harmonics.
If the higher order harmonics are not there to begin with, then the 2nd/3rd do not need to mask anything… but some might still prefer the signal modified, or distorted, so that the 2nd/3rd harmonics were present.
Some of this may also bleed into people running tube preamps into SS/Class-D amps.