>If we say 440 Hz produces a note called A, then 880 Hz, 220 Hz, 1760 Hz, 110 Hz, and so forth will also produce a note called A. An important consequence of this is that all distinct notes we could possibly come up with must exist somewhere between 440 Hz and 880 Hz. Any other pitch could be doubled or halved until it lies in that range, and thus would produce a note in that range.

That's inaccurate and backwards. 338 Hz is not in that range, and it can't be doubled or halved to fit in that, but it's a perfectly fine not. The range is more about tuning (which is a choice) than about 440-880Hz being something special.

>All of these things messily overlap and create multiple conflicting names for the same things, because they’re attempts to describe human intention rather than an objective waveform.

Nope, they don't overlap any more messily than 1+2 == 2+1, or in how 0.99999... == 1.