I'm looking at modern browser APIs and wondering why no one else is trying certain things.

getDeviceMedia and getUserMedia are very powerful these days. I haven't actually tested it but I believe a chromium browser would have no issue capturing the hw accelerated output from a game. You can pipe these media streams directly to WebRTC peers for playback on the other side. A server with a simple selective forwarding unit could enable larger scale meetings (100s of participants). All of this can happen in <1000 lines of JS and server code. Most of the heavy lifting happens in the browser engine. Concerns like automating browser permissions, global hot keys, etc. can be handled via electron or platform specific options like WebView2.

Mobile clients are a bit cursed right now. The best solution is to maintain a standard client in the app stores. Forcing everyone to sign their own mobile apps is way too much friction. And you do need native for this on mobile. Browser only / PWA has no chance in hell of providing a smooth UX on iOS or Android.