user api stops working

*This issue was originally created by [**@Joshix-1**](https://github.com/Joshix-1) at .*

### Background information

- **Dendrite version or git SHA**: 0.5.1
- **Monolith or Polylith?**: Monolith
- **SQLite3 or Postgres?**: Postgres
- **Running in Docker?**: no
- **`go version`**: 1.16.12
- **Client used (if applicable)**: - - -

### Description

- **What** is the problem: The user api stops working
- **Who** is affected: The user api, the federation api still seems to be working
- **How** is this bug manifesting: The user api doesn't respond anymore/ times out
- **When** did this first appear: I don't know; A while after starting the instance

### Steps to reproduce
It just happens all the time, every few minutes. Sometimes after half an hour.

No source files, entry points, or tests are identified. Start by reproducing the user API timeout on Dendrite 0.5.1 running as a monolith with Postgres, then inspect logs and compare the user API with the still-working federation API. Done means identifying the failure cause, restoring reliable user API responses, and adding coverage for the failure if the relevant test location can be found.

Written by the indexing model from the issue text.