Add GHA for private maintainer-only tests

[jwnimmer-tri]

In #307, I am adding a test case that helps us (as maintainer) ensure that files which are copied around inside of this repository all remain up-to-date and in sync.

After #307 merges, please add a new .github/workflows/ci.yml job which runs this test (by itself, not as part of any other existing build).

If you like, you could create a private/test/run_all.sh script for CI to call, in anticipation of us eventually adding more test programs.