[3.13] gh-120973: Fix thread-safety issues with `threading.local` (GH-121655) (#122042)

This is a small refactoring to the current design that allows us to
avoid manually iterating over threads.

This should also fix gh-118490.
(cherry picked from commit e059aa6b01030310125477c3ed1da0491020fe10)

Co-authored-by: mpage <mpage@meta.com>