I don't want to take sides in this discussion but share an anecdote. Hey, maybe even someone knows a solution for this.

I have a PC connected via on-board HDMI to a Denon AVR solely for the purpose of getting the audio to the amplifier. Windows doesn't let me use that audio interface without extending or mirroring my desktop to that HDMI port. Since there is no display connected to the AVR I don't want to extend the desktop, and mirroring heavily decreases performance of the system.

On Debian Sid the computer by default allows me to use the HDMI audio without doing anything to my desktop display. It seems the system realizes that there is no display connected to the AVR but it's still a valid sink for audio.

If I remember correctly, NVIDIA did a decent job with their on-board HDMI driver audio-wise; what brand is yours? I'm 100% the functionality is dependent on the driver rather than the OS.

Can you use optical instead of HDMI?

TOSLINK doesn't support uncompressed surround audio, while HDMI does, and I do use it.