I’d love to try and help, but even from a read-only link it’s difficult to see the areas that relate to this type of thing. I don’t want to point out excessively obvious things either, but I’d of course double check that the staging domain name you have is valid in your general settings within project settings. If it is, maybe try selecting “Unpublish” from the publish to domain drop down, then confirm it is unpublished at the URL. If it is, re-publish it, and check the URL. Maybe try changing your staging domain, maybe you tried these things already.
Could you explain further what is happening? When you click publish is it infinitely loading? Or are you finding that it shows a successful publish, but not updating at the link?
I’m sure we can find an answer for you.