Thanks for the detailed response. Let me do my best to address some of your points and explain my reasoning.
> The harmonic series is never perfectly in phase in acoustic instruments.
This is a misunderstanding. Let's take a stringed instrument as our example. When a string vibrates at a certain frequency this is called the fundamental, which is the pitch we perceive. However, there are additional frequencies that we also hear called overtones. These come from perfect subdivisions oscillating within the same string. In other words, they are in phase, because they're literally in phase. If you pluck a string at first it has a lot of energy, then it slowly decays, the string wiggles and vibrates in and out of tune, and all the while every subdivision within that string is doing the exact same thing, in phase.
> In many instruments it isn't even perfectly in tune, never mind perfectly in phase.
I'm not 100% sure what you mean by this. The Harmonic series will be derived from a single string regardless of whether it's in tune with concert A or C, that won't matter, it works at any fundamental. I think what you mean is regarding multiple strings played very closely in phase (e.g. note on a piano). This will not work unless they're relatively in tune, otherwise you'll have harmonic dissonance.
Getting back to phasing and what we perceive as a single note vs what we perceive as multiple notes. Let's consider the reciprocal arrangement and imagine a chord that consists of the same notes as the harmonics that occur in a single string. (Let's also pretend for the sake of argument that the notes are strummed with the exact same strength that their harmonic counterparts are present and that they're perfectly in tune.) Since it's a chord, we would hear, by definition, multiple pitches, not a single fundamental. The reason for this is that each note is oscillating independently, I.e. out of phase.
> In fact it isn't always in phase in electronic instruments.
So, try this. Use audacity or whatever and generate pitches according to the harmonic series and tell me what you hear. I promise you, 100%, you will hear a single note. Since a computer is generating a sine wave, and the harmonics are by definition ratios of the fundamental, they will be perfectly in phase, and you will therefore hear a single note, try it.
> None of this alters that this is a perceptual issue.
No disagreements here, and I don't think anything I'm saying is mutually exclusive with this. If you take the above example and start shifting the phasing, at a certain point you will no longer perceive the other tones as harmonics but as pitches within a chord. Our brain however will stitch them together when they're close, that's the perception aspect. Another way to think about this is with pianos. If you take my chord example from above, and make a special piano with a single hammer that strikes multiple strings tuned according to the harmonic series independently, then we will perceive a single pitch, despite them not being perfectly in phase. They're not perfectly in phase because they're actually separate strings, but our perception will stitch them together, like you said.
> Some trained musicians, piano technicians, organ makers, and so on can pick out individual overtones.
I'm not really sure what the point you're making here, if you care to clarify. We all hear individual overtones, that's called timbre. I guess people who are trained can identify them (octave, 5th, etc...), but we all have a sense of what's a bright vs dark instrument.
Edited to fix formatting issues that came from a little ascii art thing I tried to draw showing how overtones are perfect subdivisions of an oscillating string.