You're thinking too small. It's possible to have a payment infrastructure that doesn't give Visa and Mastercard money, and it's possible for it to work really well. https://en.wikipedia.org/wiki/EFTPOS
note that in practice EFTPOS is getting beaten down by still not doing contactless, while Visa Debit/Mastercard Debit contactless cards are ubiquitous in Australia, and banks pushing Visa/Mastercard Debit for the higher interchange fees.