The costs hit all customers, not just the ones who use credit cards. And the benefits accrue largely to the cardholders of premium cards, which are wealthy people. It’s a tax on being poor to subsidize the wealthy and the middlemen.
Cards should be forced into being a true commodity with perfect competition. The incentive rewards games are harmful to society.
To have competition you do need a way to differentiate, and it's going to be funded by some combination of those fees, interest/charges, kickbacks etc. Some interchange fee is simply necessary, iirc the EU attempts to prevent the egregious regressive aspects of that by capping the interchange fees of all cards. Thereby eliminating the "Amex Premia" as it were.
I reckon it's a pretty sensible step?
Remember when mobile carriers were competing via "differentiation"? Practically, that meant waiting on firmware updates for months or years, having to deal with preloaded and non-removable crapware on mobile OSes etc.
Now they're largely "dumb pipes" competing on price and quality of service, and as a consumer I couldn't be happier. (Of course, there are now different gatekeepers on our phones, but that's a different discussion.)
The clue to this is in the question. Banks offer cash-backs (and other rewards) not Visa or MC. They do so because credit cards are a commodity and they need to differentiate their product from all the other card providers.
In effect some of the fee paid by suppliers is routed back to the customers. To suppliers the cost is part of doing business, and built into the price.
As a supplier I much prefer customers to use cards over cash. Cash is very expensive to deal with in any volume. And is far more vulnerable to theft (by external people, but also staff.)