Mapman, the test is simple- put the amp on a speaker load (nearly any load will do) and run the amp up to clipping with the sine wave, and observe the results on a 'scope.
Of course others have already done that. Goto Google, click 'Images' then enter:
tube clipping characteristic
The first two images will show you the difference- the transistor characteristic being the first image. Note the squared-off waveform of the transistor test- as if someone cut off the tops and bottoms of the waveform with a knife. These sharp corners are evidence of odd ordered harmonics. This test is common to all transistor amps.
The tube amp has rounded corners- less odd-ordered and lower orders as well. This characteristic is common to all tube amps.
Now this test is at clipping but this difference between tubes and transistors has been quite well documented in the last 40 years and really isn't a topic of debate as it is so readily measured and heard. It is why nearly all guitar players use tube amp, BTW.
However, almost any audiophile will correctly argue that we don't listen to amplifiers at clipping. And I agree with you in your surmise in the second to last paragraph of your post. A cheap tube amp with lousy transformers and the like is still a crappy amp.
However, the fact of the odd-ordered harmonic distortion issue will not go away despite all this. All it says though that a transistor amplifier will sound harsh compared to a tube amp **generally speaking** (and with rare exception...).
A lot of people are OK with that, thinking that they can use a synergistic approach to deal with the harshness. But look back at my original post here- I specified the difference between sounding like *music* or sounding like a *good hifi*. Its that nuance, that nth degree, that I am talking about- not the ability to weld with the amp or the like.