Hello,
By default all users have Read Access to all spaces (this of course can be changed).
I have tried to recreate this scenario and indeed I can log in as a non-admin user and view all spaces (folders) in the system - this is normal.
But, if I try to add or create content into any space where I don't have the correct permissions I am disallowed.
Have you checked whether your non-admin user can create content in spaces where they don't have the correct permissions?
Have you also checked whether a Group that the user is a member of has any permissions on the spaces?
Thanks,
Steve