PYTHON-5349 Use drivers-evergreen-tools to start servers in GitHub Ac… #7957

Job	Run time
static	3m 40s
CPython 3.13t-ubuntu-latest	11m 12s
CPython pypy-3.10-ubuntu-latest	12m 21s
CPython 3.9-ubuntu-latest	11m 12s
Test using minimum dependencies and supported Python	35s
DocTest	1m 5s
Docs Checks	49s
Link Check	1m 42s
Test async's minimum dependencies and Python	44s
Typing Tests (3.9)	1m 7s
Typing Tests (3.11)	1m 11s
Make an sdist	15s
Install from SDist and Test	55s
	46m 48s

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