Fix RetryDestinationLimiter re-starting finished log contexts

Signed-off-by: Sean Quah <seanq@matrix.org>

Author: Sean Quah

changelog.d/12795.bugfix

@@ -0,0 +1 @@
+Fix a long-standing bug where finished log contexts would be re-started when failing to contact remote homeservers.

synapse/util/retryutils.py

@@ -16,8 +16,8 @@
 from types import TracebackType
 from typing import TYPE_CHECKING, Any, Optional, Type
 
-import synapse.logging.context
 from synapse.api.errors import CodeMessageException
+from synapse.metrics.background_process_metrics import run_as_background_process
 from synapse.storage import DataStore
 from synapse.util import Clock
 
@@ -265,4 +265,6 @@ async def store_retry_timings() -> None:
                 logger.exception("Failed to store destination_retry_timings")
 
         # we deliberately do this in the background.
-        synapse.logging.context.run_in_background(store_retry_timings)
+        run_as_background_process(
+            f"store_retry_timings-{self.destination}", store_retry_timings
+        )