[infra] Migrate some health checks to tool/ci.dart

[dcharkes]

Currently the health checks are github actions this leads to the following downsides:

1. it's hard to reproduce failures locally (you have to reverse engineer the github action and invoked packages)
2. you cannot run the same checks locally as a sanity check before submitting a PR (this is especially painful for external contributors for which CI-runs need to be approved)'
3. the versions of the tools is not pinned by the dev dependencies in your packages leading to non-hermetic builds.

However, writing code manually per repo will likely lead to code duplication. So may we should provide a Dart API for the health checks that users can add as a dev dependency. WDYT @mosuem?