gh-104050: Run mypy on `clinic.py` in CI (#104421)

* Add basic mypy workflow to CI
* Make the type check pass

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Co-authored-by: Erlend E. Aasland <erlend.aasland@protonmail.com>
Co-authored-by: Nikita Sobolev <mail@sobolevn.me>
Co-authored-by: Hugo van Kemenade <hugovk@users.noreply.github.com>