Blutooth PXC 550-II as Headset on Windows 10 : Cannot activate "Handsfree telephony" because "The parameter is incorrect"

My PXC 550-II used to work just fine as a headset - until this morning where Windows just wouldn't capture any sound from its microphone (as per the little indicator next to the corresponding entry in Sound Settings Recording). It works just fine with my phone though, both as headphones and as headset. I now have unpaired it from my PC, restarted my headset, and paired with it again, and now there is no related entry any more in Recording , nor is there an entry for the headset mode in Playback (there is however an entry for my device as Headphones i.e. for audio playback only, and it works as

Question by Charles on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.