gh-114271: Clean Up _threadmodule.c

[ericsnowcurrently]

This strictly moves code around into sections and adjust some names, in the interest of making it obvious where to find code and where to add new code. This is partly motivated by gh-114839. There will be follow-up PRs in the near future that take advantage of this cleanup.

At the moment, this is mostly a demonstration of some of the cleanup that I think would help make gh-114839 easier to review. I'd actually split this up into two or three parts. My gut also tells me that there's probably one extra PR that could be split out of gh-114839, though I haven't quite identified what it would be.

CC @mpage

• Issue: Make _threadmodule.c thread-safe in --disable-gil builds #114271