Create a mypy-primer like workflow in CI for Sphinx

Is your feature request related to a problem? Please describe.

When e.g. adding new warnings, it is good to know what the impact is likely to be. In the static typing ecosystem, @hauntsaninja's mypy-primer is used automatically in CI to report on many popular open source projects (including Sphinx!). I think we should explore doing something similar for Sphinx, as an aide to pre-release testing and to give us confidence when adding new warnings. For example, I don't currently have a good estimate for #13400 and if we could immediately make it a warning (cc @jayaddison).

Describe the solution you'd like

I don't know how much / if there is anything we could take from mypy-primer as scaffolding here. @hauntsaninja may be able to advise (hope you don't mind the ping!). Ideally we'd have something in CI that we could run, either nightly / on request / on every PR. We'd need to capture the environment setup for building docs for each chosen project. Probably isolated environments? We'd then need to choose a set of initial OSS projects to test against.

Describe alternatives you've considered

Doing nothing.

Additional context

• https://github.com/sphinx-doc/sphinx/issues/12242#issuecomment-2065914075 (cc @picnixz)