Initial `/sync` when filtered to a single room can be very slow

This issue has been migrated from [#10842](https://github.com/matrix-org/synapse/issues/10842).

---

### Description

The `/rooms/{roomId}/initialSync` endpoint is deprecated and the [documentation](https://spec.matrix.org/unstable/client-server-api/#get_matrixclientr0roomsroomidinitialsync) suggests using `/sync` instead.

However, using `/sync` with a room filter has drastically worse performance for users in very many rooms,
eg. taking up to 400 seconds for a user in 2,200 rooms and pulling 100,000+ events from the database.

### Steps to reproduce

```
GET /_matrix/client/r0/sync?filter=%7B%22room%22:+%7B%22rooms%22:+%5B%22!XXXXXXXXXXXXXXXXXX:matrix.org%22%5D,+%22state%22:+%7B%22rooms%22:+%5B%22!XXXXXXXXXXXXXXXXXX:matrix.org%22%5D%7D,+%22timeline%22:+%7B%22rooms%22:+%5B%22!XXXXXXXXXXXXXXXXXX:matrix.org%22%5D,+%22senders%22:+%5B%22@xxxxxxxxxx:matrix.org%22%5D%7D,+%22ephemeral%22:+%7B%22limit%22:+0,+%22types%22:+%5B%5D%7D,+%22account_data%22:+%7B%22limit%22:+0,+%22types%22:+%5B%5D%7D%7D,+%22presence%22:+%7B%22limit%22:+0,+%22types%22:+%5B%5D%7D,+%22account_data%22:+%7B%22limit%22:+0,+%22types%22:+%5B%5D%7D%7D
```

### Version information

- **Homeserver**: matrix.org

Start by reproducing the filtered GET /_matrix/client/r0/sync request described in the issue and profile the initial-sync path that handles it. Trace why a single-room filter reads 100,000+ events for users in many rooms; done means the filtered initial /sync no longer exhibits the reported excessive latency and database reads.

Written by the indexing model from the issue text.