Generally old dependencies shouldn’t be a problem, if everything needed is pinned. As long as the Python used can still be installed. (That might stop working eventually, GitHub - actions/setup-python: Set up your GitHub Actions workflow with a specific version of Python has been dropping Python versions in the past.)
But as I said: if publishing the Ansible 9 docs fails, doesn’t also publishing the ansible-core 2.16 docs likely fails? And these still need to be published, I guess? So this isn’t a problem specific to Ansible docs, but to ansible-core docs as well?
(And yes, we likely need to do at least some regular checks whether the tooling still works - not updating it is OK as long as it still works.)