After some investigation, seems like
pulseaudio-module-bluetooth is missing. These are the steps to install it:
$ sudo apt install -y pulseaudio-module-bluetooth
$ sudo pactl load-module module-bluetooth-discover
$ sudo service bluetooth restart
This should be enough. No need for a reboot.
Not all headsets allow simultaneous hi-fi output sink and input source, i.e.: there are different profiles (a2sp, hsp, hfp) which may or may not be supported simultaneously. The "problem" is that a single Bluetooth channel has not sufficient band for both communications to occour simultaneously. Only "dual channel" headsets support different profiles simultaneously.
I cannot tell for sure, but my Philips SHB9850NC was not able to play well with HSP profile: the microphone was simply not working. I was not able to figure out what the reason was. I've tried to switch profiles and I've tried lots of things for a fair amount of time. I cannot claim that "it does not work": maybe it was just my plain incompetence to make it work on Linux... but it shouldn't be rocket science, right?
I've ended up buying another headset (a dual channel) which comes with a pre-paired USB dongue which supports multiple profiles simultaneously.
If you are looking for a headset for videoconferencing over WebRTC for example, you'd better try brands/models targetting this specific market, advertising very clearly this specific usage on call centers or other business applications.
Thanks @Hannu for pointing out a script which helps select device profiles:
Enabling the equalizer: