I would actually like to hear libertarians explain away this one.
I'll give it a shot.
The payment processors are propped up by the government giving companies additional powers they do not/should not hold (Patriot Act deputizing payment processors, as Null pointed out in his big post about this topic)
Regulation via government means that it is impossible for myself or another private citizen to even try to begin to create a new payment processing organization or company, regulation ensures this is prohibitive in cost and not a worthwhile investment (If a billionaire could start their own Visa or Mastercard, don't you think they would do so?)
Cryptocurrencies were/are the solution to this, because the idea was that you would no longer have to rely on these processors and you can engage in payment via crypto in order to make the transaction both peer-to-peer (not subject to scrutiny) and non-taxable. Obviously this would not stand, because all governments/states want to control what you are able to buy, and they want their cut. More and more crypto regulations are being rolled out, and sites where crypto was popularized for buying real goods (Silkroad) have been seized.
The Libertarian perspective is that Visa/Mastercard is propped up by the state and no one is allowed to compete with them. This is not a Free market, because the government will continue to work on their behalf, because Visa/Mastercard will intervene on the behalf of the state, and their duopoly only secure because of this.
The ideal would either be to dismantle this and deregulate the industry, allowing there to be hundreds of payment processor who compete with each other, or the further proliferation of cryptocurrencies and utilizing them for day to day transactions.