If anything, that's an argument in favour of the whole thing being government controlled -- it's clearly a natural monopoly with a large capital expenditure requirement and a comparatively minimal (as a percentage of transactions) ongoing charge required to continue to run it. I mean, it clearly doesn't cost Visa / MC any more to make a £/$/€45 000 payment than a £/$/€5 one – database writes are not cheaper or more expensive if the fixed point value you change is smaller or larger (subject to fitting within the range requirements, etc)...
Brazil has gone that route with PIX [0]. It's a fund transfer system with realtime confirmation. Many e-commerce stores accept it besides debit/credit, they can confirm the payment almost instantaneously and they usually give discounts if you use PIX instead of debit/credit (3-10%).
Also, paying at stores, clinics, doctors, etc.. all done with PIX nowadays. Very few people are carrying cash these days. Very few clinics had a payment machine to accept debit/credit cards and now with PIX they have no incentive to do so anymore.
To transfer funds, you can specify the user's ID (which can be a phone number, national tax identification or some random string the user creates). To pay for purchases, you usually scan a QR code or copy a payment string using your bank's app/website. It's very convenient.
> If anything, that's an argument in favour of the whole thing being government controlled
Absolutely, and I kinda expect this to become the case eventually across Europe, as the "cashless society" vision becomes inevitable. If you make it de-facto compulsory for citizens to operate electronically, it's only fair that there should be the sort of guarantees and accountability that only the State can provide, even if you lose a bit of innovation or efficiency.
