Perhaps what we need is an amp with variable second and third harmonic controls, so that we can dial in the exact mix that we prefer.
This can be done with digital signal processing, or something similar. I think its a good idea - that is I think what Nelson Pass used on test subjects to reach his optimal tweaking of included harmonics.
I don't know how difficult it would be to realistically implement in an analogue amp intended for home use.. Probably would be expensive - just a wild guess.