Fix lost logcontext on HomeServer.shutdown()

[MadLittleMods]

Fix lost logcontext on HomeServer.shutdown()

Same fix as #19090

Spawning from working on clean tenant deprovisioning in the Synapse Pro for small hosts project (https://github.com/element-hq/synapse-small-hosts/pull/204).

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]

This is the main fix!

See the Deferred callbacks section of our logcontext docs for more info (specifically using solution 2).

Heads-up, I wrote the docs too so it's my assumptions/understanding all the way down. Apply your own scrutiny.