I have exactly the same issue. I can, however, rent movies via my ATV app on my phone and then watch them on the ATV by finding the rented movie in “recent purchases”.
Credit card authorisation is successful via the QR an iPhone browser but then the ATV does not react - just stays on the screen with the QR code. Sometimes the “auth with your phone” option generates the message on the phone but when I do that, a message appears on ATV saying the card was not authorised. Very weird!
I have reset the ATV to factory settings, signed out of the store and and also completely out of everything and signed back in but no difference.