Modernization of asyncio infrastructure / Deprecate Tornado

[fjetter]

This is an issue to track efforts towards some modernization efforts of our asyncio infrastructure.

Very roughly this is about

• Addressing all warnings triggered by py3.10
• Slowly migrating away from tornado
• Deal with anti-patterns around our usage of futures, tasks and callbacks, e.g. for better exception handling

Related issues

• Enable error warning mode #6048
• Remove pytest-asyncio #6050
• Remove deprecated usage of tornado.IOLoop.run_sync #6049
• Remove tornado loop kwarg in Server #6051