Logging loop after SIGHUP when a `StreamHandler` is configured

This issue has been migrated from [#6550](https://github.com/matrix-org/synapse/issues/6550).

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If you use the default logging config, which has a `StreamHandler` pointing at sys.stderr, and do not set `--no-redirect-stdio`, then when you send synapse a SIGHUP, it will stack overflow and abort.

The problem appears to be that when you SIGHUP it, the streamhandler gets repointed at the redirected stderr, which goes back into a logger: so as soon as anything is logged, we get an infinite loop.

Reproduce with the default logging config and a SIGHUP while leaving --no-redirect-stdio unset; trace how the StreamHandler pointing at sys.stderr is repointed to redirected stderr. Confirm the issue is resolved when SIGHUP no longer causes recursive logging, a stack overflow, or an abort.

Written by the indexing model from the issue text.