Some merchants reject cards with higher fees; i.e., Amex is not accepted at some merchants with lower margins (i.e., grocery stores).

It would be impractical for merchants to accept some branded cards and not others. Imagine "we accept "Chase Premium One" card, but not "Chase American Airlines" card." Very confusing for consumers. If it's a whole category, like Amex, it's easier to refuse it (besides, low income consumers are unlikely to have an Amex card).