The customers using high reward cards would then soon discover that they are just paying for those rewards themselves, and the free market would quickly push everyone to the most efficient payment systems.
At least in the US, the contract the retailer signs with the CC processor to be able to accept CC's explicitly bans passing the card fees on to the customer as a specific line item in the receipt.
Which really translates to the merchant increaseing their prices by 1.02x to compensate for the 2% (avg. estimate) fee the cards charge. So the customer still pays, but they do not directly see what part of the price goes directly to the payment network.
But they are still barred from doing the reverse (line item for X% more for CC transaction on the receipt).
Eventually we'll just clamor for merchants to just put a single price tag on everything and not worry us about how much of each item is going to all of their various costs.
Edit: If you really believe the freemarket is the best to determine what is the preferred mode of exchange. Then decentralized currencies are inevitably going to win out, especially if they're equally accessable. (Meaning no need for extensive tech knowledge or expensive mining rigs).
A currency that has a built in UBI and virtually no transaction costs is going to be attractive to consumers. And merchants will eventually gravitate towards it more as they see it enables consumers to consume more.