Error 503 suggests the Plex Media Server is still running but most likely deadlocked and requests were just queued up until they reached few thousands at which point it starts returning 503 errors to requests
Deadlocks can only be investigated through getting a forced process dump and debug server logs and where possible to get a list of connections before forcing a dump - the server should in theory allow the specific requests for /connections even when deadlocked - lets switch your problem to this forum topic Server not accessible - Error 503 - #3 by sa2000
The linked post shows the diagnostics needed for deadlocks
