[3.13] gh-141004: Add a CI job ensuring that new C APIs include documentation (GH-142102)

(cherry picked from commit 1a7824a)

Co-authored-by: Peter Bierma <zintensitydev@gmail.com>
Co-authored-by: Hugo van Kemenade <1324225+hugovk@users.noreply.github.com>

Author: ZeroIntensity