Mastercare and visas are generally issued by government-owned companies or banks in each country.
Please give an example of a government-owned company that issues those cards! I'm all ears!
Oh and just because you're a teacher at some school and you get your wage on a special card that doesn't mean the school or the minister of education is neither the issuer nor the balance holder. Most government agencies are not licensed for financial or money transmitting licenses, they won't even be able to do such a thing.
Considering how "huge" those two companies are, I am %100 sure that we are going to end up with a lot of trouble if we end up with anything else then these companies figuring out a way. I mean they will probably have hard time if we just use our apps, because banks would be able to just directly deal with it, without needing cards.
The app is just an extension!
What matters is the settling thing, not what Visa does between you and the bank but what Visa does between your bank and the merchant's bank!
That's the infrastructure that matters and nothing has even come close in two decades to be a rival for it!