Remove unused `start_background_update`

This was only used in a unit test, so let's just inline it in the test.

synapse/storage/background_updates.py

@@ -400,27 +400,6 @@ class BackgroundUpdater(object):
 
         self.register_background_update_handler(update_name, updater)
 
-    def start_background_update(self, update_name, progress):
-        """Starts a background update running.
-
-        Args:
-            update_name: The update to set running.
-            progress: The initial state of the progress of the update.
-
-        Returns:
-            A deferred that completes once the task has been added to the
-            queue.
-        """
-        # Clear the background update queue so that we will pick up the new
-        # task on the next iteration of do_background_update.
-        self._background_update_queue = []
-        progress_json = json.dumps(progress)
-
-        return self.db.simple_insert(
-            "background_updates",
-            {"update_name": update_name, "progress_json": progress_json},
-        )
-
     def _end_background_update(self, update_name):
         """Removes a completed background update task from the queue.
 

tests/storage/test_background_update.py

@@ -25,12 +25,20 @@ class BackgroundUpdateTestCase(unittest.HomeserverTestCase):
         # the target runtime for each bg update
         target_background_update_duration_ms = 50000
 
+        store = self.hs.get_datastore()
+        self.get_success(
+            store.db.simple_insert(
+                "background_updates",
+                values={"update_name": "test_update", "progress_json": '{"my_key": 1}'},
+            )
+        )
+
         # first step: make a bit of progress
         @defer.inlineCallbacks
         def update(progress, count):
             yield self.clock.sleep((count * duration_ms) / 1000)
             progress = {"my_key": progress["my_key"] + 1}
-            yield self.hs.get_datastore().db.runInteraction(
+            yield store.db.runInteraction(
                 "update_progress",
                 self.updates._background_update_progress_txn,
                 "test_update",
@@ -39,10 +47,6 @@ class BackgroundUpdateTestCase(unittest.HomeserverTestCase):
             return count
 
         self.update_handler.side_effect = update
-
-        self.get_success(
-            self.updates.start_background_update("test_update", {"my_key": 1})
-        )
         self.update_handler.reset_mock()
         res = self.get_success(
             self.updates.do_next_background_update(