Race condition with room creation means room directory lacks read-after-write consistency in monolith mode

This issue has been migrated from [#15526](https://github.com/matrix-org/synapse/issues/15526).

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It's possible to query the room directory after the room creation and not see the room.

### Reproduction steps

1. Using monolith Synapse
1. Create a public room called `foo` with `/createRoom`
1. Immediately after seeing a successful create room response, search the room directory for the room (`/publicRooms` with `{ filter: { generic_search_term: 'foo' } }`)
- It's important to use search because it [bypasses the cache](https://github.com/matrix-org/synapse/blob/0e8aa2a1b28dfce374294450a015d18884c89d36/synapse/handlers/room_list.py#L98-L109)
1. Notice that the room is missing from the response ❌

If you give the homeserver some time, the room will show up.

### Notes

This regressed in https://github.com/matrix-org/synapse/pull/15229 where I guess the timing changed just enough for this to be possible. You can see the full bisect, investigation and context for how I noticed in https://github.com/matrix-org/matrix-public-archive/pull/208#discussion_r1183294630

It seems like we're `await`'ing everywhere so not sure why this would happen. And seems weird that it would happen in monolith mode since there is not replication lag and we're awaiting everything.