PYTHON-5763 Improve async test coverage #9110

Job	Run time
DocTest	1m 1s
Coverage	12m 18s
CPython 3.13t-ubuntu-latest	11m 37s
CPython 3.10-ubuntu-latest	11m 39s
Docs Checks	39s
CPython pypy-3.11-ubuntu-latest	12m 42s
Link Check	3m 41s
Make an sdist	21s
Typing Tests (3.10)	47s
Test minimum dependencies and Python	56s
Typing Tests (3.11)	46s
Integration Tests	1m 6s
static	2m 3s
Install from SDist and Test	1m 13s
	1h 0m 49s

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