Asianometry provides a great summary as to how both of them came to be: For Visa, a 1976 rebranding of the BankAmericard program. For Mastercard, a 1966 meeting of banks as opposition to BankAmericard.
It seems the consensus is that a taxes are only bad if you have to pay the government. If it's a small set of companies that collectively own a virtual monopoly, it's because they earned it.
Payments themselves are not a technical challenge, no matter who's doing it. The fundamentals are trivial. You move numbers between accounts.
It's tackling fraud and dealing with disputes that's a challenge.
Consider that the largest payment card network on Earth (China UnionPay, 7 billion cards) - decided it was easier just to bootstrap acceptance in the US by a partnership with Discover rather than connecting directly to merchants.
If you want a new scheme to work, distribute something like social security and welfare cheques through it. That immediately forces broad acceptance.