Alright, that actually did work.
In the admin page, it now tells me a bunch of scripts have been changed and of course that I should run setup.
When I do run setup, the website throws an http 500. I can still log into the backend, but of course the message remains.
debug.log doesn't show any setup related entries, only that zpFormattedDate is deprecated.
setup.log actually is not updated when I click the button in the backend.
I checked in /zp-core/setup/ and all files are .xxx and haven't been changed since I pressed the button in the backend.
I **am** running MySQL 5.7 still. Is that why setup fails? Shouldn't it at least unprotect the setup files before?
Since I am in deep anyway, I will try to rectify issues now and not leave the instance in a "broken" state if I can manage that.
/edit: going to /zp-core/setup.php and running that has the same behaviour of course. And if I can't figure that stuff out, I will probably set up a new instance on the side and just migrate over more slowly now