Revert "Move start_doing_background_updates() to SynapseHomeServer.start_background_tasks() (#19036)"

[MadLittleMods]

Revert "Move start_doing_background_updates() to SynapseHomeServer.start_background_tasks() (#19036)"

Why

See #19036 (comment)

Revert while I figure out the tests in #19057

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]

Merged as the change we're reverting shouldn't find it's way into an RC (release candidate) that will most likely happen tomorrow.