Fixes this happening on staging:
[network] 2024-01-29 17:48:24 [verbose]: WebSocketNotificationsClientConnector connected!
[network] 2024-01-29 17:48:24 [verbose]: WebSocketNotificationsClientConnector connect attempt failed: Can't use WaitSafely from inside an async operation.
I'm not sure how I ever allowed that `.WaitSafely()` to be there. It
did feel rather dangerous but then I must have forgotten and never
noticed it failing. Which is weird because you'd think that would be
caught by testing that chat isn't working but I'm pretty sure that I
tested that chat *was* indeed working.
Anyway now that entire flow is replaced by something that should
hopefully be somewhat more sane? It has a whole retry flow with logging
now which should be more robust than what was there previously (failing
to start to listen to chat events killing the entire websocket
connection for very little good reason).
This is a prerequisite for https://github.com/ppy/osu/pull/25480.
The `WebSocketNotificationsClient` was tightly coupled to chat specifics
making it difficult to use in the second factor verification flow.
This commit's goal is to separate the websocket connection and message
handling concerns from specific chat logic concerns.
