[CI] Simplify pushing Docker images for tooling repos

[ErikSchierboom]

We've recently simplified the way Docker images are pushed for the tooling repos: https://github.com/exercism/csharp-test-runner/blob/master/.github/workflows/docker.yml This issue helps keep track of the status of the various PRs to add this new workflow to the track tooling repos.

[iHiD]

@ErikSchierboom Shall we keep this open as a tracking issue?

[ErikSchierboom]

@iHiD Definitely! There are still quite a lot of them that are still open.