I am afraid I am not willing to wait until someone considers to do something. I need to use my PC from now on.
It just happened that the Intel graphics+audio driver vanished completely. Not in any software I can select HDMI as output. There's is just a remote device availabe instead of the HDMI before. De- and re-installing didn't help.
The HD audio driver is listed in the device manager though:
location 3: 126.96.36.19997 intel display audio (intel igpu)
Any further ideas which help short-term? Despite abandoning Intel graphics?