Hi there Jens,
How awesome of you to share this. The bad news is, it did not solve the issue. The good news is, thanks to you the issue is solved anyway. I'll explain why.
Your response triggered me to dive deeper than I did so far. What I did sofar was comparing 2 Joomla sites, both same Joomla version, same SPPB version but different Joomshaper templates. Where 1 did work with PHP8.0, the other not. So my conclusion was: it's the template.
However, I did not have a look at actual error messages. So I turned on the development mode and found out 2 error messages.
Where one was related to a file which I assume is part of the core: /%root%/plugins/system/gzip/helper.php. I was able to fix the error (replacing {} with [] around some parameter) and that solved the actual issue. The other error was not blocking the usage of PHP8.0 at all.
So this case can be closed as the template did not seem to cause any issue at all. My appologies. But thanks for your response which was in the end the trigger to find the actual issue :-)
Thanks,
Louis