[3.12] gh-124309: Modernize the `staggered_race` implementation to support e… (#124574)

gh-124309: Modernize the `staggered_race` implementation to support eager task factories (#124390)

Co-authored-by: Thomas Grainger <tagrain@gmail.com>
Co-authored-by: Jelle Zijlstra <jelle.zijlstra@gmail.com>
Co-authored-by: Carol Willing <carolcode@willingconsulting.com>
Co-authored-by: Kumar Aditya <kumaraditya@python.org>
(cherry picked from commit de929f353c413459834a2a37b2d9b0240673d874)

Co-authored-by: Peter Bierma <zintensitydev@gmail.com>