Mastercard and Visa take very little out of the transaction compared to the interchange fee that goes to the issuing bank. I believe the issuing bank would take $2.50 on a $100 dollar transaction vs $0.10 for Visa and Mastercard roughly.
In europe it's more like $0.35 on a $100 transaction for standard personal Mastercards and Visa due to regulation, the rest goes to the issuing bank to cover their costs and card rewards, and to your merchant bank to cover their costs and risk of chargeback. In US and Asia the interchange fees are much higher due to lack of regulation, thus it's not unusual for small business to pay a bundled rate of upwards of 4% if little credit history
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.
