> The money extracted by the credit card companies and Visa causes merchants to raise prices for everyone regardless of whether they have a rewards card or use a credit card at all.

This creates in effect a massive money transfer from the poor, who do not use rewards cards, to the rich consumers who do.

Not quite. Credit card companies obligate merchants to charge the same prices regardless of whether you pay with a card, but merchants frequently don't honor that obligation. And there are also merchants who only take cash.

The poorest customers are likely to patronize these merchants. They're also likely to be given discounts that aren't card-related; the whole idea of price discrimination is that, because impoverished customers have low willingness to pay, you charge them less.

In a voluntary system, money transfers are always going to end up being much smaller than they looked like they would be when you thought about their effects, because people adjust their behavior to avoid them.