Refactor and convert `Linearizer` to async (#12357)

Refactor and convert `Linearizer` to async. This makes a `Linearizer`
cancellation bug easier to fix.

Also refactor to use an async context manager, which eliminates an
unlikely footgun where code that doesn't immediately use the context
manager could forget to release the lock.

Signed-off-by: Sean Quah <seanq@element.io>