PYTHON-5672 Refactor connection checkout to reduce layers #4477

Job	Run time
Make SDist	50s
Build wheels for win32	4m 22s
Build wheels for win_amd6	3m 59s
Build wheels for manylinux_ppc64le	14m 13s
Build wheels for manylinux_i686	2m 44s
Build wheels for macos	4m 19s
Build wheels for manylinux_s390x	13m 50s
Build wheels for win_arm64	6m 18s
Build wheels for manylinux_x86_64	2m 5s
Build wheels for manylinux_aarch64	14m 18s
Download Wheels	7s
	1h 7m 5s

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