PYTHON-5672 Refactor connection checkout to reduce layers #4473

Job	Run time
Build wheels for macos	3m 41s
Build wheels for manylinux_aarch64	14m 50s
Build wheels for win32	5m 7s
Build wheels for win_amd6	5m 22s
Build wheels for manylinux_i686	2m 30s
Build wheels for manylinux_ppc64le	13m 46s
Build wheels for manylinux_x86_64	2m 4s
Build wheels for win_arm64	6m 21s
Make SDist	52s
Build wheels for manylinux_s390x	15m 13s
Download Wheels	7s
	1h 9m 53s

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