And make every website part of a botnet? I sure hope not.

WebSockets are already a problem because many web devs don't know that other origins can connect to them and potentially fetch data they're not supposed to (even ignoring the fact you can construct these without a browser), unleashing raw sockets to the web platform would be hell.

WebRTC and WebSockets provide more than enough already. If you need even more, your browser just isn't the place for this type of code IMO.