Hi @trentmu,
thanks for your input, I had not checked $ bench clear-cache. Unfortunately, it has no affect on the issue.
I have also followed the logic in [ERROR] Products page "missing or moved" - #9 by seteq to no success. Seems like this happens otherwise.
As I can set up a fresh system which behaves normally (products visible without login) I am ruling out a bug per se and try to find the configuration glitch…
I have looked through the Website Generator for Item but could not find an access restriction (or hint to configuration)…
Again, thanks to you both for your support!
Here a picture showing the issue (left with login, right in private mode/no login):
