So digging through the conformity stuff listed here
https://www.tesla.com/ownersmanual/models/en_us/GUID-DAD2ACC3-FBA3-4474-AF11-EF052D4322FC.html
Under "CarPC" is YZP-ATC5CPC001 which appears to be an LG bluetooth module which I believe is integrated into the MCU on these cars.
But there's also a device listed as "BT USB hub", which appears to be the USB-C hub for the front
External Photos External Photos -- I suspect this might be what's being used for the bluetooth headsets. Notably that FCC submission was not made until May 2021, so it's very possible that the early USB hub just didn't have bluetooth.
If that's the case, installing 1656787-00-B is probably what you need. Also notably, I could have sworn that the Model S catalog used to list 1755200-00-B as the USB hub (which is currently still listed as the data+power hub in the model 3/y catalog). I'm guessing that's a variant without bluetooth and what they used to install in the S.
Edit: So I tried unplugging that module on my 2023 S, and with it unplugged, my car no longer picks up my headphones. Plugging it back in, things work as expected.
Without having a 2021 car to play with, my bet is that module is the key