Is there no interest in having retropie run as a service on top of a full desktop environment?
Not on the Raspberry Pi.
RetroPie (ES + emulators) works fine on a PC platform from the desktop, but the audio configuration is handled by the system/desktop environment and there's not much that RetroPie configures at this point, it just used the default audio device that's register in the system (i.e. in PipeWire/PulseAudio).
EDIT: as explained before, you need to find the right Audio Card setting which works for you. Try using PCM/Master instead of HDMI, it should work.