Merchants pay a bit more in fees than banks give in rewards, but definitely not 2% more.
I think AMEX makes about 65% of its revenue on interchange fees?
Amex charges higher fees (many of their cards charge the consumer directly, and their merchant fees are higher) and don't charge interest on many cards (charge cards). Their business model is different.
The marginal cost of servicing another person when they already have a consumer base of people with debt is very small (close to zero). I'm trying to say it's still profitable, but not where they make the bulk of the revenue.