[3.14] gh-133146: Add the old public `get_value` method to documentation and refactor code. (GH-133301) (#142110)

also uses it within the internals in a few places.
(cherry picked from commit db098a475a47b16d25c88d95dbcf0c6572c68576)

Co-authored-by: Duprat <yduprat@gmail.com>