The old documentation said that to unpublish a GitHub Pages site, you had to delete the branch that was the publishing source of the Pages site. If using a user site and publishing from the master branch, you first had to set another branch as the default before deleting master. (I’ve linked to an old version of GitHub Enterprise documentation because I couldn’t find an old version of Github documentation, but the instructions were similar).
I tried that and it worked. The site was unpublished and I didn’t have to rename the repo. I don’t know if this will be disabled in the future (so you really can’t unpublish a Pages site unless you delete or rename the repo), or if it’s the documentation that doesn’t cover it, but I thought I should leave a comment since you mentioned that you’ll notify the documentation team to make some changes.