gh-124309: Revert eager task factory fix to prevent breaking downstream (#124810)

* Revert "GH-124639: add back loop param to staggered_race (#124700)"

This reverts commit e0a41a5dd12cb6e9277b05abebac5c70be684dd7.

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

This reverts commit de929f353c413459834a2a37b2d9b0240673d874.