It would not be easy (or reasonable) to take a public build like that private to build a 'secret' image version, and I'm not sure GitHub even has the capability to do that. I would not trust a build pushed to Docker Hub separate from the established / public / auditable process they have in place.
All dependent (official) images need to go through the same process, so that makes it fairly problematic, and I think Docker needs to take a look at it for the future.
It should definitely have a 'high impact' workflow that bypasses some of those manual steps.