To my knowledge the network and the bank take roughly a 50/50 split of the transaction fee. Visa and Mastercard are the ones with market control, so you wouldn't expect them to get only a tiny slice.
Visa and MasterCard began as consortiums of banks, so they made the system in banks' favors. Good rewards cards pay 2% cash back on all purchases, while transaction fees on rewards cards are in the 2-3% range, which necessitates that most of that fee go to the issuing bank, or the rewards programs would be money losers... they're not.
Actually Visa's Credit Assessment rate is .14%. They raised their prices by 1bpt in January 2019 and wrote themselves a $500 Million check by doing so. This is one of the reasons why Visa's stock performs so well over time.
I can't remember the exact figure, but I believe there is roughly only a two basis point spread between what the network providers take versus the operating costs to run the network. Therefore, any "disruptor" trying to enter would have to burn cash to not only build out the network but to also lower prices to try and steal market share. And the result would really only be 1 BPS saved on each transaction for the consumer.
