The problem is network effects. If you accept Visa or Mastercard then you lose business, because many customers have only one or the other. So you need both, which means you can't play them off each other for a better deal and they aren't actually competing with each other.
Combine this with contract terms in many places preventing merchants from passing on fees and it gets even worse, because card networks can gain customers and increase their network size by offering rewards and then use the network effect to raise fees. If the merchant can't charge more to customers whose card network charges them higher fees, the cardholder has the incentive to use the card with the highest rewards (and thus the highest fees), so you get a ratchet of increasing processing fees despite an overall trend of falling underlying processing costs due to automation.
Combine this with contract terms in many places preventing merchants from passing on fees and it gets even worse, because card networks can gain customers and increase their network size by offering rewards and then use the network effect to raise fees. If the merchant can't charge more to customers whose card network charges them higher fees, the cardholder has the incentive to use the card with the highest rewards (and thus the highest fees), so you get a ratchet of increasing processing fees despite an overall trend of falling underlying processing costs due to automation.