Moonlight/Sunshine developer here, having all of these people praising the work done by us is a wonderful thing. We developers work very hard to make Sunshine faster, smoother and more powerful each day. So thank you!
I'm a contributor for the Sunshine project: https://github.com/SunshineStream/Sunshine project, which allows Nvidia GameStream to be used on Linux systems with Moonlight clients. I haven't tested the Linux side of things, but it has many ways to both encode the screen or acquire it (via KMS, Pipewire or x11grab IIRC).
Personally I managed to run Outline using a standard OIDC software like Keycloak. While more troublesome it should work just fine without external tools
> Moonlight expects: Each client connects to start their own private game session
Nope, it's a Wolf design choice, eg. Sunshine allows users to concorrenly connect to the same instance/game