Clean up some LoggingContext stuff (#7120)

* Pull Sentinel out of LoggingContext

... and drop a few unnecessary references to it

* Factor out LoggingContext.current_context

move `current_context` and `set_context` out to top-level functions.

Mostly this means that I can more easily trace what's actually referring to
LoggingContext, but I think it's generally neater.

* move copy-to-parent into `stop`

this really just makes `start` and `stop` more symetric. It also means that it
behaves correctly if you manually `set_log_context` rather than using the
context manager.

* Replace `LoggingContext.alive` with `finished`

Turn `alive` into `finished` and make it a bit better defined.

docs/log_contexts.md

@@ -29,14 +29,13 @@ from synapse.logging import context         # omitted from future snippets
 def handle_request(request_id):
     request_context = context.LoggingContext()
 
-    calling_context = context.LoggingContext.current_context()
-    context.LoggingContext.set_current_context(request_context)
+    calling_context = context.set_current_context(request_context)
     try:
         request_context.request = request_id
         do_request_handling()
         logger.debug("finished")
     finally:
-        context.LoggingContext.set_current_context(calling_context)
+        context.set_current_context(calling_context)
 
 def do_request_handling():
     logger.debug("phew")  # this will be logged against request_id