Log saturation prevents Synapse from working

This issue has been migrated from [#16711](https://github.com/matrix-org/synapse/issues/16711).

---

### Description

I'm using the Element service attached to Synapse and I've noticed that when the synapse logs are saturated, the Element service stops working.

### Steps to reproduce

- Connection to Element not working
- Synapse service log dump
- Functional Element connection

### Homeserver

another homeserver

### Synapse Version

1.89

### Installation Method

pip (from PyPI)

### Database

single PostgreSQL server

### Workers

Single process

### Platform

Synapse runs in a Python virtual environment, under python 3.9, within a Linux Centos7 virtual machine.

### Configuration

_No response_

### Relevant log output

```shell
no relevant log output
```

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

_No response_