Processing fees aren't inherently expensively. They're low in Europe, and high in the US only because they go to rewards programs. You're paying 2% more but getting 2% back. Legislation could easily regulate them to eliminate rewards programs and bring them back down low.
Chargebacks are supposed to be expensive because they're a deterrent to businesses acting in ways that will lead customers to attempt chargebacks. And then they usually have an element of manual review which costs $ as well. Chargebacks being expensive is a feature not a bug. And they're an easy consumer protection option that isn't even a possibility with cash.
Chargebacks exist because the element of agency in payment is broken in the electronic money systems we've created. When using a card, you don't give merchants your money, you provide them with enough information to take your money away from you.
The UX of those two things is similar enough that nobody really cares, until it bites them in the ass, that is.
Chargebacks are supposed to be expensive because they're a deterrent to businesses acting in ways that will lead customers to attempt chargebacks. And then they usually have an element of manual review which costs $ as well. Chargebacks being expensive is a feature not a bug. And they're an easy consumer protection option that isn't even a possibility with cash.