https://wiki.archlinux.org/title/PipeWire check the applications in GUI section.
You can likely save the wiring or script it to reset, it may also depend on your desktop manager or other app that assigns what app uses what audio output.
There is a roundabout way to do this, and Pipewire has a graphical way to do this, so you can run it all natively on system audio.
In essence you create a new audio sink, and reroute your game audio to the sink while also routing the sink back to your speakers/headphones, so you can hear it.
Then in OBS you can pick that game only sink as your audio source. Nothing else would get captured.
But I’ll check that plugin in other comments, seems like a more robust solution for OBS specifically.
I’m looking at tuwunel to setup instead of synapse as soon as it’s ready. Still matrix, but maybe better for small instances.