Make the frontend available sooner (Part 1 of 2) (#36263)

* Part 1 of 2 (no breaking changes in part 1).

When integrations configured via the UI block startup or fail to start,
the webserver can remain offline which make it is impossible
to recover without manually changing files in
.storage since the UI is not available.

This change is the foundation that part 2 will build on
and enable a listener to start the webserver when the frontend
is finished loading.

Frontend Changes (home-assistant/frontend#6068)

* Address review comments

* bump timeout to 1800s, adjust comment

* bump timeout to 4h

* remove timeout failsafe

* and the test

homeassistant/components/websocket_api/http.py

@@ -165,7 +165,9 @@ class WebSocketHandler:
             EVENT_HOMEASSISTANT_STOP, handle_hass_stop
         )
 
-        self._writer_task = self.hass.async_create_task(self._writer())
+        # As the webserver is now started before the start
+        # event we do not want to block for websocket responses
+        self._writer_task = asyncio.create_task(self._writer())
 
         auth = AuthPhase(self._logger, self.hass, self._send_message, request)
         connection = None