PYTHON-5538 Clean up uv lock file handling #8133

Job	Run time
CPython 3.9-ubuntu-latest	11m 36s
static	4m 23s
CPython 3.13t-ubuntu-latest	11m 42s
CPython pypy-3.10-ubuntu-latest	12m 31s
DocTest	1m 8s
Docs Checks	38s
Typing Tests (3.9)	1m 10s
Typing Tests (3.11)	1m 0s
Test using minimum dependencies and supported Python	39s
Link Check	7m 34s
Make an sdist	13s
Test async's minimum dependencies and Python	42s
Install from SDist and Test	55s
	54m 11s

Provide feedback