Square and similar platforms charge the same rate across all card networks for ease of use - I suspect lower volume and a variety of cards sort of balance things out. It also makes things easier to say "we take everything and it doesn't matter."
Amex has been working with some merchants/payment processors to try to encourage growth (through their OptBlue initiative):
https://optblue.com/
Finally, higher tiers like Visa Signature, Visa Infinite, World Mastercard, and World Elite Mastercard often have higher fees for merchants with varied payment processing rates (like most larger retailers where "Amex is higher"). These have crept up to even be more than Amex in some cases, which is part of the reason why Kroger has been fighting with Visa in some markets.