I finally got around to fixing this issue and as I mentioned before I wanted to give you some info on it.
Surprisingly all I had to do was install the child theme and activate it. That didn’t mess up the layout of the site and I could update the parent theme afterwards without issues. So now the site is finally back in working order. And I’m only left to wonder why updating the parent theme if you have no child theme set up messes up the site, while installing a child theme with the latest version and switching over to it seems to not affect it at all.
But yeah, this info might help some of your other customers. It sure would have helped me if I had known that this problem could be solved so easily. But all I’ve ever heard from you guys is that I will lose my customizations if I install a child theme, so I have to copy everything over manually, etc.