Hello again!
I'm pretty sure that this is not a local issue (i.e. one where the settings on your device matter) because everyone here has the same problem (iTunes and App Store deny access to OS-X 10.8 and older, and there might be other services which are blocked for which one needs to connect to security-sensitive servers such as appleID.apple.com).
Therefore I wouldn't recommend wasting time right now with resetting my device or other time- and nerve-consuming tasks until this issue has been addressed by Apple (I hope they will do so).
I have been trying to find out if there is anything that Windows Vista and XP have in common with OS-X 10.8 and older, since it is known (to me as of today) that Windows XP and Vista lost access to the iTunes Store May 25, 2018, and since it looks like OS-X 10.8 and older are getting the same treatment after that deadline...
Other than being outdated in the form of terminated support by Microsoft for Vista and XP, I could not come up with any technical similarities, but maybe the OS-X 10.8 and older with the Safari version 6.x.x. is simply not up to par anymore security-/encryption-wise in a similar way.
The problem for me is that, in the current state, I could not upgrade even if I wanted to. Nobody informed me of this although my device is registered with Apple. Nothing popped up on my desktop. There was, as far as I recall, no warning that OS-X 10.8 and older would be affected as well by the changes introduced on May 25, 2018 (which I am pretty sure are the reason for all this), or that I would not be able to upgrade after May 2018. I do however remember the months, or years, of warning of the terminated support for Windows XP.
Just around Xmas I updated my Dad's iPad 2 from iOS 4 or whatever was on there when I bought it for him, to iOS 9... the device is 7 years old! This is not possible in the current situation. That can't be right, and that is why I am hopeful that there will be a solution. Hopefully not USB sticks for everyone affected to boot from and forced upgrades to High Sierra which might be a little too much for my MBP?
Good luck 🙂