I have had a similiar idea some time ago but way less automated and windows-only. Will see if I can contribute anything useful.
Slightly OT: I found CloudRetro and this project some time ago when dabbling in WebRTC and looking at Pion. Interestingly I had a very similiar idea back then and even managed to create a working prototype. Basically cloud-gaming-multiplayer on a single browser page, including 4 low-latency streams for the 4 players and one slightly higher-latency stream for "spectators" (using OvenMediaEngine).
It currently only supports gamepads but allows you do join/leave the game at any time, or leave and let a spectator have a go.
I'd like to open source it sometime soon but currently its barely functional. I'm using a brokenly modified version of the OBS FFMpeg plugin to pump video and audio over RTP to Pion. OBS likes to crash randomly now (due to me not knowing what I'm doing in regards to changing codec parameters etc). Also I still have issues with still frames when there is minimal movement (h264) or the frame freezing and disintegrating into a pixelated mess (using vp8).
Of course the OBS setup requires a bit more setup (and I believe there is no headless-functionality for it?). I'm using a spare windows machine to to stream the games (usually Mario Kart Double Dash (Gamecube)). I even tried a short session in VR with others in BigScreen.
Just throwing some ideas for others out I guess.