[3.12] Convert `doc.yml` workflow to be reusable (GH-103914 + GH-105151) (#107042)

Co-authored-by: Sviatoslav Sydorenko <wk.cvs.github@sydorenko.org.ua>
Co-authored-by: Adam Turner <9087854+AA-Turner@users.noreply.github.com>
Co-authored-by: Hugo van Kemenade <hugovk@users.noreply.github.com>

(cherry picked from commit 88d14da76f579fe014cbd7c15e42be4234135fe9)
(cherry picked from commit eaa670228066220f08c8d73f80365c50058d40b8)