Optimize `/backfill` to respond faster

[matrixbot]

This issue has been migrated from #15653.

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Since /backfill is a big part of /messages performance, optimizing /backfill would help other servers respond to our /backfill requests faster which means we can respond to /messages faster.

From the examples below, it seems like we have these obvious opportunities to improve:

• get_backfill_events(...): Seems like a pretty slow query that we could probably turn into a recursive query.
• get_current_state -> get_filtered_current_state_ids: Takes ~200ms
• is_host_joined: uncached call took ~200ms
• _get_state_groups_from_groups(...): A known slow thing
  • Previous optimizations:
    • Optimize filter_events_for_client for faster /messages - v2 matrix-org/synapse#14527
    • Re-use work of getting state for a given state_group (_get_state_groups_from_groups) matrix-org/synapse#15617

3.45s /backfill?limit=100 request

934ms /backfill?limit=100 request