Wait for response to be sent

[MadLittleMods]

Spawning from #18804 (comment) as mentioned by @richvdh,

[...] the goal is just to use async/await all the way up the chain so the trace_servlet doesn't close until we're finished and LoggingContext.scope sticks around until the end as expected.

But we're no longer using LoggingContext.scope since moving to OpenTracing's ContextVarsScopeManager in #18849 (reverted in #18849)

The benefit now is TODO

---

In terms of tracing, TODO

Before	After
	.

Dev notes

You can see how I did the simpler implementation of async HTTP resources in https://github.com/element-hq/synapse-small-hosts/pull/313

Pull Request Checklist

• Pull request is based on the develop branch
• Pull request includes a changelog file. The entry should:
  • Be a short description of your change which makes sense to users. "Fixed a bug that prevented receiving messages from other servers." instead of "Moved X method from EventStore to EventWorkerStore.".
  • Use markdown where necessary, mostly for code blocks.
  • End with either a period (.) or an exclamation mark (!).
  • Start with a capital letter.
  • Feel free to credit yourself, by adding a sentence "Contributed by @github_username." or "Contributed by [Your Name]." to the end of the entry.
• Code style is correct (run the linters)

[MadLittleMods]

Feel like we might want to copy this comment up the chain to respond_with_json and _send_response/_send_error_response