It's how they pay for their best-in-class rewards and offerings. This means card holders are more likely to make big purchases on an Amex than other cards, and why many small ma n' pa shops don't accept them.
In my country, they offer some of the best rewards rates and customer care services. Their no-frills chargeback protections and airport lounge network are somewhat better than what I have with my Visa card. However, Amex lacks offline acceptance and can be quite challenging to meet the "milestones" for annual fee waiver.
Other answers are good but keep in mind that other brands can be pretty expensive. If you are B2B but most of your business is using a credit card, Visa and Mastercard business card and government purchase card transactions are more costly than American Express.
I don't think Stripe loses any money on a transaction. All interchange rates are less than 2.99%, but some are very close. For very low volume/low ticket price, stripe is a pretty good deal. But if high volume, there's bound to be a better option that pays off considerably the sooner you implement it.
Now, if you can get most of your customers using debit cards or ACH-type transactions, you can really achieve a low cost if you use a processor with interchange-based pricing.
Why support American Express? Their members tend to be better customers in my experience, and they appreciate that you support their preference.
In my experience people with American Express also carry a Visa or Mastercard, so there’s limited downside to refusing to accept it.
Crazy that we live in a world where 3% is “a good deal”. In Europe the fees are capped at 0.15%, and I’m sure even that is well above the true costs of payment processing. Interchange fees are such a clear case of market failure that I’m surprised they haven’t been regulated.
I agree, and the rewards system gives consumers a false idea that they are getting a net benefit. The new game is charging a 3.99% service charge to the customer to pay with card, which takes the pressure off the vendor and allows the processor to take a larger cut. In some places, like tourist traps, you don't have any other options. The ATM in town might charge $3.99 withdrawal fees as well.
I was told by an insider that the reason why emv+pin is not used in the US is because Visa/MC actually profit from fraud. If fraud were cut considerably, you could argue that the network is only providing 5-10¢ of value per transaction, and cut their rates with legislation.
If we went to a system like PIX in Brazil, and dropped rewards, we could do away with most of the industry. People would only use credit cards if they needed credit.