Let's approach this logically.
First things first. If you come in here like gangbusters, having all the appearances of a troll, you're not going to get anywhere. This site is constantly under siege by trolls who gleefully take advantage of Amir's policy of tolerance, and many members here have long ago gotten sick and tired of them. Knock off the hyperbole and ask questions rather than adding fuel to flame, and this thread may turn into an interesting discussion.
A large headroom figure is not a measure of how BAD the power supply is, it is the measure of how uncontrolled the power supply is. Although many regard a loosely controlled power supply as "bad", that doesn't have to be true. Designers and manufacturers continually balance costs against characteristics under load, and make what they believe are the best decisions for their products in the real world.
You said:
It is technically impossible for all things to be absolutely equal between two amplifiers that have different headroom characteristics. The differences will always be obvious on the bench. But that's not the problem with your statement (above). Your statement discussed the
sound of the amplifiers being compared, and you used the adverb "always", but you didn't specify the exact conditions of "equality".
If your basis of equality is RMS measured power, then amp #1 would output (let's say) 100 watts continuous without any headroom. Amp #2 would output 100 watts continuous, but have 3dB headroom, for a short-term output of 200 watts. Under the condition that the 3dB of headroom would be necessary to trace the signal, amp #2 would actually sound
better, not worse.
If your basis of equality is the total power envelope, then you would compare (let's say) amp #1 with 100 watts continuous and no headroom with amp #2, which would now
necessarily be defined (becasue we've moved the goal posts) as an amp with 50 watts output continuous and 3 dB headroom for 100 watts total. So you would be comparing what would seem to be a 50-watt amp to a 100-watt amp. Under that circumstance, there could be an audible difference ... but in my experience it would be slight. IMO, it would favor the 100-watt amp.
This is assuming that both amps are designed to deliver their power into extremely difficult loads as seen in some hard-to-drive speakers such as electrostatics, or dynamic loads with low impedance combined with odd phase angles.
Unfortunately, that doesn't happen often. Manufacturers who favor high-headroom designs usually do so for reasons of economy. They want to offer their customers greater short-term capabilities while lowering the demands on the overall circuitry so as to cut costs. Such designs are not usually created to deal with the exceptionally demanding speaker loads that are the minority on the market. Instead, they are created to deal advantageously with common loads as seen in the vast majority of loudspeakers on the market.
I said, "
unfortunately, that doesn't happen very often." In reality, I am not sure that it happens at all ... ever. I'm not sure that it CAN happen ... at least, not if the amps are both rated exactly the same and under the same loads.
Which brings up the last factor; advertising.
If the FTC rule (as originally implemented) were adhered to rigidly, differences in circuit design capability would be stated ( and obvious) in the specs. But almost no one does that anymore.
So comparing the power of two amps on an equal basis depends on the point at which an advertiser wishes to set the limit. 1% distortion? Beginning of the "knee"? 4 ohms at the expense of 8 ohms?
If the advertiser wanted to set the rating dependant on short-term headroom limits, then I'm sure that you are correct; there would be
some circumstances under which that amp, like amp #2 above, would sound inferior to an amp, like amp #1, with a higher RMS figure.
Whether that would be encountered in the normal 1-watt-average-output scenario of the average home is up for grabs, so to speak. And the cost-cutting designers know that. They bet on it to work in their favor.
So your use of the word "always" is inaccurate and inappropriate. Your use of the word "sounds" as the basis for your comparison is, I believe, untenable. It may apply in a rare case, but the sound difference cannot be determined to be verifiable in each and every case.
As usual, I might be wrong.
Jim