user_left_room distributor signal is unreliable

[matrixbot]

This issue has been migrated from #13673.

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In Synapse, there is a class called Distributor, which maintains a list of callbacks for a named signal and calls them all when the signal is fired: https://github.com/matrix-org/synapse/blob/develop/synapse/util/distributor.py#L43

There's only one signal defined currently and it's user_left_room, which is used in two places:

1. DeviceHandler, to mark a remote user's device list as unsubscribed
   https://github.com/matrix-org/synapse/blob/1a209efdb2a6c51e52dd277de7581099852877ae/synapse/handlers/device.py#L569-L575
2. TypingWriterHandler, to stop a user's typing indicator when they leave a room.
   https://github.com/matrix-org/synapse/blob/1a209efdb2a6c51e52dd277de7581099852877ae/synapse/handlers/typing.py#L311-L315

Unfortunately, the user_left_room signal is currently only reliable when local users leave a room normally (and not via a ban!).
This manifested as #13651 for device lists. Once #13651 is fixed, device list tracking will no longer use the signal.

We should look into whether the typing listener is necessary and if it can be moved elsewhere, so that we can eliminate this bit of architecture that's only used by one tiny part of synapse.