Unable to open too many DevTools pages due to browser per-host connection limit being exhausted by SSE connections

Moving this from https://github.com/Dart-Code/Dart-Code/issues/5229.

If you run DevTools and open ~5 tabs for it, you'll find you can't open any more. This appears to be because the SSE connections are tying up the maximum number of connections Chrome allows to a single host.

From my testing of opening many https://echo.websocket.org/.ws pages, I _think_ WebSockets are not counted towards this limit, so the solution may be to migrate the SSE connections to WebSockets. However, I don't know if there may be internal issues with WebSockets that led to SSE being used initially (@kenzieschmoll do you know? there's currently a mix of WS + SSE, so I presume WS must work everywhere now?).

Another option may be to use HTTP2 (so requests can share connections), but it's not clear to me if Dart has a HTTP2 server (I found https://github.com/dart-lang/http2 but it doesn't look like a complete solution?).