Appservice stream position got stuck ~10k behind current, preventing requests to appservices

This issue has been migrated from [#13950](https://github.com/matrix-org/synapse/issues/13950).

---

### Description

A repeat of https://github.com/matrix-org/synapse/issues/1834 essentially


### Steps to reproduce

Unclear - it suddenly became sad.

### Homeserver

t2bot.io

### Synapse Version

1.68.0 + custom patches

### Installation Method

pip (from PyPI)

### Platform

Ubuntu physical hardware.

### Relevant log output

```shell
Available upon request.
```

### Anything else that would be useful to know?

After manually fastforwarding the stream position and restarting the worker it appeared to be running about 3-5 minutes behind for longer than expected. This may have been due to a larger server restart causing caches to be evicted during peak hours, though.

Start by comparing this report with the referenced issue #1834 and review the stream-position behaviour around the appservice worker. The only reproduction evidence is the reported lag, manual fast-forward, restart, and subsequent 3–5 minute delay; use relevant logs if they can be obtained. Done means the appservice stream no longer falls far behind current or prevents requests.

Written by the indexing model from the issue text.