"Or maybe you can do it externally, by varying the mouth cavity. That will weakly couple to the instrument, since the mouth acts like a Helmholtz resonator itself (it has an audible effect with the recorder)"

mmm, a bit of a change of stance now from your previous

"I can't imagine why you would want to invoke Helmholtz resonators as components of a flute."

:-)
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"controlling breath pressure - you don't want to compromise that by messing about with embouchure. "

Yep, that's just something that people who play embouchure flutes have to do all the time to 'blow the notes in' (called 'lipping in' and 'lipping out') ....

The thing that amazes me most about all physical musical instruments is just how 'messy' they are, how much effort in construction and performance technique has to be employed to 'control' them ('wolf tones', etc). real world things do not care about mankind's efforts to think up nice simple uncomplicated theories of how they work - they just do what they damn well please! It is the player that is 'doing magic', imposing their will on a chaotic universe ...
:-0

The only 'musical instruments' that are in contrast 'simple and clean and totally predictable' are the electronically generated instruments that use a master oscillator and fixed 2:1 gated dividers to produce the sub octaves - there is even a dedicated chip to produce within a few cents a full octave of root 12 tempered notes.

If you have ever played a real pipe organ with those hundreds, even thousands of pipes, some of them 16/32 or even 64 foot long, then you do feel that these 'generated instruments' are somewhat 'sterile' and fixed ...

Robin - multi-instrumentalist and once trained when young many years ago in orchestral conducting and arranging (which means that you must understand how each instrument works, what it can do, what it can't, its pitch and volume and timbre range, how they work together, etc) ...