This depends on the card. The cards people associated most with Amex (platinum, gold, etc) are charge cards and you're expected to pay them in full every month instead of carrying a balance. That's part of why Amex historically had higher fees for retailers. Those fees were their only source of income.

I'm sure most banks have good enough financial modelling that they can tweak their credit card approvals to get whatever balance between retail fees, annual fees, and interest they want for that card.