PYTHON-5530 Reduce usage of legacy test runner (#2642) #8648

Job	Run time
Docs Checks	40s
Link Check	1m 39s
static	2m 50s
Integration Tests	40s
CPython 3.13t-ubuntu-latest	9m 16s
CPython pypy-3.11-ubuntu-latest	9m 16s
Typing Tests (3.10)	1m 5s
DocTest	50s
CPython 3.10-ubuntu-latest	9m 16s
Typing Tests (3.11)	1m 5s
Test minimum dependencies and Python	46s
Make an sdist	10s
Install from SDist and Test	1m 3s
	38m 36s

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