Increase timeout waiting for Core API to work around 2025.12.x issues (#6404)

* Fix too short timeouts for Synology NAS 

With Home Assistant Core 2025.12.x updates available the STARTUP_API_RESPONSE_TIMEOUT that HA supervisor is willing to wait (before assuming a startup failure and rolling back the entire core update) seems to be too low on not-so-beefy hosts. The problem has been seen on Synology NAS machines running Home Assistant on the side (like in my case). I have doubled the timeout from 3 to 6 minutes and the upgrade to Core 2025.12.1 works on my Synology DS723+. My update took 4min 56s -- hence the timeout increase was proven necessary.

* Fix tests for increased API Timeout

* Increase the timeout to 10 minutes

* Increase the timeout in tests

---------

Co-authored-by: Jan Čermák <sairon@users.noreply.github.com>

supervisor/homeassistant/core.py

@@ -48,7 +48,7 @@ _LOGGER: logging.Logger = logging.getLogger(__name__)
 
 SECONDS_BETWEEN_API_CHECKS: Final[int] = 5
 # Core Stage 1 and some wiggle room
-STARTUP_API_RESPONSE_TIMEOUT: Final[timedelta] = timedelta(minutes=3)
+STARTUP_API_RESPONSE_TIMEOUT: Final[timedelta] = timedelta(minutes=10)
 # All stages plus event start timeout and some wiggle rooom
 STARTUP_API_CHECK_RUNNING_TIMEOUT: Final[timedelta] = timedelta(minutes=15)
 # While database migration is running, the timeout will be extended

tests/homeassistant/test_core.py

@@ -387,7 +387,7 @@ async def test_api_check_timeout(
         ):
             await coresys.homeassistant.core.start()
 
-    assert coresys.homeassistant.api.get_api_state.call_count == 3
+    assert coresys.homeassistant.api.get_api_state.call_count == 10
     assert (
         "No Home Assistant Core response, assuming a fatal startup error" in caplog.text
     )