PYTHON-5519 Clean up uv handling (#2510) #8076

Job	Run time
CPython 3.13t-ubuntu-latest	11m 43s
CPython 3.9-ubuntu-latest	11m 38s
static	4m 8s
Link Check	5m 57s
CPython pypy-3.10-ubuntu-latest	12m 27s
Typing Tests (3.11)	1m 2s
DocTest	1m 2s
Typing Tests (3.9)	1m 4s
Make an sdist	12s
Test async's minimum dependencies and Python	40s
Test using minimum dependencies and supported Python	34s
Docs Checks	41s
Install from SDist and Test	1m 3s
	52m 11s

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