Add type checker to `examples/client` #3999

Job	Run time
checks / readme-snippets	11s
checks / pre-commit	32s
checks / test (3.11, lowest-direct, ubuntu-latest)	45s
checks / test (3.10, lowest-direct, ubuntu-latest)	45s
checks / test (3.12, lowest-direct, ubuntu-latest)	50s
checks / test (3.12, highest, ubuntu-latest)	49s
checks / test (3.13, lowest-direct, ubuntu-latest)	43s
checks / test (3.10, highest, ubuntu-latest)	47s
checks / test (3.10, lowest-direct, windows-latest)	2m 24s
checks / test (3.13, highest, windows-latest)	2m 31s
checks / test (3.11, highest, ubuntu-latest)	45s
checks / test (3.11, lowest-direct, windows-latest)	2m 33s
checks / test (3.13, lowest-direct, windows-latest)	2m 26s
checks / test (3.13, highest, ubuntu-latest)	43s
checks / test (3.12, lowest-direct, windows-latest)	3m 42s
checks / test (3.11, highest, windows-latest)	2m 27s
checks / test (3.12, highest, windows-latest)	3m 22s
checks / test (3.10, highest, windows-latest)	2m 27s
	28m 42s

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