I still don't know how to unpair on some black boxes. My partner has a speaker, and I think auto-pairing occurs if used previously, so we have a strange ritual of turning off bluetooth on other devices until it seems to correct itself when wanting to use it.
I've always assumed this was a UI issue.
I've had phones that have presented themselves as a pointing device to a computer. Some as modems. Some that have transferred/pushed files okay on some OSs, not others. I have no idea what's going on with Bluetooth most of the time, and it's quite infuriating. On Linux Blueman doesn't give much away. On a Windows PC I have, using Bluetooth kills WIFI, and requires a reboot to fix.
From a mobile perspective if I say try and transfer a file from a phone, it can turn Bluetooth on, fail and then leave it on.
Wireless headphones sound like a great idea though! But again not sure which protocols are being used. Again that's a UI issue, I've a streaming speaker - and have no idea about the stream quality of what I'm listening to. Oh for some simple feedback diagnostics.