[Docs] Publish to site should fail if docgen does

See [this action run](https://github.com/beyond-all-reason/spring/actions/runs/14276569071/job/40020490723).

Note emmylua_doc_cli has failed to generate docs from the templates. This caused all documentation to be removed from the normal site display.

If we can detect docgen to fail, we shouldn't publish site. Better older documentation than none

Start with the GitHub Actions workflow used by the linked action run and trace how emmylua_doc_cli reports failure before the site is published. Done means a documentation-generation failure prevents publishing and leaves the previous site documentation available.

Written by the indexing model from the issue text.