Resolve and caches paths for CachingStaticResource in the executor

[bdraco]

This is part of an optimization cycle to improve the responsiveness of the UI during first pickup and startup

Proposed change

Noticed in #74443 that we do I/O in the event loop

aiohttp does the stat in the executor so we likely should as well https://github.com/aio-libs/aiohttp/blob/6013bd4e17ae46e796e28c1380ff3bde2161008d/aiohttp/web_fileresponse.py#L143

Since this handler is specifically for static paths, we can also cache the result of the executor job.

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[probot-home-assistant]

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[bdraco]

I should move lru-dict into base requirements

[bdraco]

Manual testing is good on this one, but I want to add some more coverage before merging