[3.11] gh-106368: Increase Argument Clinic CLI test coverage (GH-107156) (#107190)

Instead of hacking into the Clinic class, use the Argument Clinic tool
to run the ClinicExternalTest test suite.

(cherry picked from commit 83a2837b328c58b243f7d97bec12c64ec66681c5)

Co-authored-by: Erlend E. Aasland <erlend@python.org>
Co-authored-by: Nikita Sobolev <mail@sobolevn.me>