It's not that the distortion signature is too small, it's that no one measures it. No SPL meter is going to be better than ±0.1db @ 85dB. It likely has a nonlinear response. In absolute terms, it's not even close.
It gets measured alright. But I think I see what the confusion is. An SPL meter is only useful if the speaker is making sound, whereas we've been talking about measured output power. The two are not the same, owing to tube amps having a slightly higher output impedance in most cases- that will cause minor FR errors. My prior statement in this regard is correct:
at any given frequency, for a given amount of power into a given load (8 Ohms for example) the voltage and current will be the same on account on the power formula (1 Watt= 1 Amp times 1 Volt).
Back in the 80's Bob Carver tweaked an amp to sound exactly like another: The Carver Challenge | Stereophile.com
I remember that. He failed too; a 35dB match was all he could get.
FWIW, we gave up on matched level listening tests in the 70's simply because it's about as worthless reading the spec sheet.
If one amp was making more higher ordered harmonics, it likely would have sounded louder. An SPL meter gets to the bottom of that pretty quickly.
Don't conflate the spec sheets with the measurements. Spec sheets are IMO the Emperor's New Clothes. Measurements are not.