[3.12] gh-101524: Only Use Public C-API in the _xxsubinterpreters Module (gh-105258) (gh-107303)

The _xxsubinterpreters module was meant to only use public API.  Some internal C-API usage snuck in over the last few years (e.g. gh-28969).  This fixes that.
(cherry picked from commit e6373c0d8b59512aa7f0dea7f3fb162b6ed10fa4)

Co-authored-by: Eric Snow <ericsnowcurrently@gmail.com>