Use AnyIO with `fast_acquire` synchronization

[MarkusSintonen]

Summary

Related to https://github.com/agronholm/anyio/pull/761 and https://github.com/encode/httpcore/pull/922#issuecomment-2325139599

This uses AnyIOs fast_acquire-mode in its synchronization primitives (instead of replacing AnyIO with vanilla asyncio). Which gives a big performance boost (see above comment lint).

Checklist

• [x] I understand that this PR may be closed in case there was no previous discussion. (This doesn't apply to typos!)
• [x] I've added a test for each change that was introduced, and I tried as much as possible to make a single atomic change.
• [x] I've updated the documentation accordingly.