Run test webserver within main async loop

[Archmonger]

Description

The test webserver struggles to startup/shutdown on GitHub Actions, seemingly due to weirdness with Python threads.

Unfortunately, subprocesses won't work due to pickling issues. As a result, the only obvious alternative is running the webserver directly within the main thread's asyncio loop.

This seems to work for the purpose of running tests.

Checklist

Please update this checklist as you complete each item:

• Tests have been developed for bug fixes or new functionality.
• The changelog has been updated, if necessary.
• Documentation has been updated, if necessary.
• GitHub Issues closed by this PR have been linked.

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