Use AnyIO instead of pytest-asyncio for running async tests

[agronholm]

This is the first step to implement #166.

[codspeed-hq]

CodSpeed Performance Report

Merging #242 will not alter performance

_{Comparing agronholm:test-with-anyio (073f086) with main (fa8f9b5)}

Summary

✅ 14 untouched

[agronholm]

The linting failure is not due to my changes, it exists in main already.

[Cito]

Thanks for the contribution, @agronholm. Will review later. Never mind any linting issues, will fix them.