AmEx, at least classically (though less so today) is a “charge card”, not a “credit card”. It’s a subtle distinction, since they do extend short-term credit, but until Pay Over Time became a thing recently, your AmEx bill had to be paid in full every month.

As they made no money from balances carried, they have traditionally had higher merchant fees and annual fees for cardholders. The flip side, for the merchant, is that wealthy American cardholders vastly prefer using AmEx, so if those are part of your target market, you are much better off accepting it.