`FederationDeniedError` is not a `SynapseError`

This issue has been migrated from [#14190](https://github.com/matrix-org/synapse/pull/14190).

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`FederationDeniedError` is not a `SynapseError`.

Basically the same reasoning that @erikjohnston gave in https://github.com/matrix-org/synapse/pull/13816#discussion_r993262622

> I'm not sure it makes sense for this to inherit from `SynapseError`. The risk is that this manages to bubble all the way to the client API, returning a 403, which seems entirely wrong.

See https://github.com/matrix-org/synapse/pull/14191 where `SynapseError` is given more context and clarified.

### Todo

- [ ] Make sure any upstream callers handle `FederationDeniedError` properly and not just the inherited `SynapseError` class

### Pull Request Checklist

* [ ] Pull request is based on the develop branch
* [ ] Pull request includes a [changelog file](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#changelog). The entry should:
- Be a short description of your change which makes sense to users. "Fixed a bug that prevented receiving messages from other servers." instead of "Moved X method from `EventStore` to `EventWorkerStore`.".
- Use markdown where necessary, mostly for `code blocks`.
- End with either a period (.) or an exclamation mark (!).
- Start with a capital letter.
- Feel free to credit yourself, by adding a sentence "Contributed by @github_username." or "Contributed by [Your Name]." to the end of the entry.
* [ ] Pull request includes a [sign off](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#sign-off)
* [ ] [Code style](https://matrix-org.github.io/synapse/latest/code_style.html) is correct
(run the [linters](https://matrix-org.github.io/synapse/latest/development/contributing_guide.html#run-the-linters))

Start by tracing upstream callers of FederationDeniedError and compare their handling with SynapseError. Check that the exception cannot incorrectly bubble to the client API as a 403, and verify that each caller handles FederationDeniedError explicitly. Run the relevant tests after updating the callers.

Written by the indexing model from the issue text.