Banks are already processing payments. When you put your credit card in your bank's ATM, the ATM is managed by the bank. It connects to your bank server and all the authorisation and processing is being done by the bank. Visa and mastercard provide connection to other banks. When you put your card in another bank's ATM or buy something in a business associated with another bank.

Getting ride of the Visa/Mastercard duopoly would mean that all banks must connect to all other banks, which they do not want to do, believe me. That would be an administrative hell for them. Which is why they've been putting up with those two for so long. These are the google of banks. They are convenient.

The alternative would be some sort of joint venture between all the banks. The result would less fees for the banks but no benefits for the client as the bank would keep the margin and it would still be a privacy hell.

Another alternative is a public utility. But a lot of people will be as uncomfortable providing their payment data to the government.