The problem is not that credit card companies charge large fees. The problem is that they do not allow to pass the fee on the customers. Because of this I don't like European regulation - instead of capping the fees they should just make clauses that not allow merchants to set a card acceptance fees, illegal.

The clause that doesn't allow passing fees on the customer is the only thing that makes market non-free.

I think it would be only fair if people paying with a card were charged more. They get the cash back from the bank anyway.

Also I know that there are super-discounted stores in my country that do not accept cards for this reason.

They absolutely allow you to pass the fee on to the customers... as long as you phrase it as a cash discount from the posted price, instead of a credit card fee on top of the posted price.