Docs: GitHub Actions site publication scripts need to be updated [Progress] In Review

The current GitHub Pages deploy scripts do not seem to work anymore, probably because of changes in GitHub. We should use updated scripts.

Note: This is based on team experience at our company. We have working scripts based on the current scripts and GitHub's static publishing scripts. If this issue is approved we will create an MR with the updated scripts that we currently use (so we know they're OK).