When it's free in the EU, then you'll have a point. Until then you're not addressing his argument.
It's supposed to handle the problem, but we all know that the companies just increase prices to compensate.
I don't know the real number for the economy as a whole, but the experience I do have (mostly in the higher risk/cost "card-not-present" scenarios) actual transaction costs are always higher than they seem at first glance.
For example, retailers bear the majority of fraud risk. Add that in, and the cost of anti-fraud.. it adds up. Then there's the consumer side, where users "pay" fees, pay interest rates..
My point is that all these digital payment (including the paper-originally CCs) tend to become bottlenecks and oligopolies. Banks, CCs, gateways.. all tend to centralisation, regulation and low competition. All the ingredients for price inflation.
The idea of having to pay a fee to a series of private companies in order to be permitted to use money is absurd.
Dealing with cash isn't free to merchants either. And that cost is passed on to the customer.
Sure, governments that are pushing for a cashless society have ulterior motives. But you can't blame them for capping fees that used to be just spread onto everybody, regardless of the payment method. Now at least they're supposed to be comparable.
Banks have gotten more and more greedy, and come up with new and creative ways to charge extravagant fees, and thus charge merchants for cash orders and deposits (even though it is free for private customers).
These fees are however a very different concept, as the bank here is only a storage service. The fees are entirely avoidable by buying a safe, and the fees are not related to money changing hands.
With a credit card, fees are unavoidable and part of the transaction itself, and they're not even going to your bank, meaning that they want a slice of the cake from elsewhere (despite them technically earning money on your stored money through investments).