Beating and intermodulation distortion are entirely different things. They look similar on an oscilloscope, but they're not and they don't sound the same.
>Instruments are not limited to 20-20kHz. They can have overtones well above this range.
Correct. You can't hear the overtones beyond the upper portion of the hearing range (many people believe you can).
>In a real listening environment (i.e. live performance) these overtones have a chance to interact with one another in the air.
In reality they do not unless you're driving the air so hard the trough rarification is approaching hard vacuum. (That's not actually impossible. It's how ultrasonic audio 'beaming' devices work). Some performances are powerful enough to get close, eg, if you're sitting six feet from the pipe organ.
Once you're driving air so hard it becomes nonlinear, thus introducing intermodulation distortion in the air, that distortion produces actual audible-range distortion products. And because the distortion you're hearing is in the audible range, a recording will sample and reproduce it accurately.
You're hearing the audible _result_ of IMD, you're not somehow listening to the distortion curve itself.
> It is possible that these overtones may beat with one another
You're continuing to confuse beats and IMD, but here you're talking about beat frequencies, so Yes. But beat frequencies are a sort of auditory illusion. If one of the frequencies that would produce a beat is inaudible--- there's no beat. Easy to test, go try it.
> and cause inter-modulation products in the audible range.
IMD is not a beat. Inaudible ultrasonics will produce audible artifacts when the underlying reproduction system is nonlinear (another way of saying 'there's intermodulation distortion'). However, that's a playback artifact. If the IMD products were audible in the original signal, audible range sampling would reproduce them.
If it wasn't audible in the original performance, it should not be part of the recording, and it should not be part of the playback.