I think a lot of people don't realize it's possible to use UDP in browsers today with WebRTC DataChannel. I have a demo of multiplayer Quake III using peer-to-peer UDP here: https://thelongestyard.link/ Direct sockets will have their uses for compatibility with existing applications, but it's possible to do almost any kind of networking you want on the web if you control both sides of the connection.
Not really peer to peer though, is it? The q3 server is just running in the browser session that shares a URL with everyone else?
It won't work completely offline unfortunately, as the server is required for the connection establishment step in WebRTC. A peer-to-peer protocol for connection establishment on offline LANs would be awesome, but understandably low priority for browsers. The feature set of WebRTC is basically "whatever Google Meet needs" and then maybe a couple other things if you're lucky.