If (a) everyone who owned a credit card actually had the means to pay for the good/service with available cash (even if not literally in their pocket) then (b) credit cards wouldn't be necessary and thus (c) good/services that typically use credit cards would be ~1-3% cheaper. In other words, the people who are getting rebates are artificially getting them b/c if credit cards didn't exist, all goods/services would be cheaper.
* someone needs to drive to the bank to ensure the register has enough pennies, quarters, and dollar bills to make proper change in the morning
* a system needs to be developed to prevent employees from stealing cash
* to avoid being a crime target, there needs to be a way to safely store larger amounts of cash
* there's a cost of an occasional fake bill being accepted by an employee
* someone needs to drive to the bank to deposit the day's earnings into the bank account
It's not like the US has zero businesses that deal exclusively in cash, so there should be success stories of disruptive startups going against the establishment by foregoing merchant fees and passing the savings on to consumers.
But outside of Arco am/pm gas station, I struggle to find a good example where a cash-only business, or chain, or a sector would offer consistently lower prices than their credit-card-accepting counterparts.
Most US businesses don't bother with offering discounts for cash/debit payments, so it's not like cash-only lifestyle will generate dramatic savings.