PYTHON-5767 Implement DRIVERS-3427 phase 1 rollout changes #9122

Job	Run time
Typing Tests (3.10)	1m 3s
Coverage	57s
static	2m 59s
DocTest	57s
Make an sdist	17s
CPython pypy-3.11-ubuntu-latest	56s
CPython 3.13t-ubuntu-latest	49s
Docs Checks	39s
Integration Tests	58s
Typing Tests (3.11)	1m 13s
CPython 3.10-ubuntu-latest	49s
Test minimum dependencies and Python	52s
Link Check	1m 39s
Install from SDist and Test	1m 14s
	15m 22s

Provide feedback