That's going to depend on market dynamics between buyers and sellers.
The long version is, read: https://en.wikipedia.org/wiki/Tax_incidence and replace "government" and "tax" with "VISA" and "transaction fees".
To be honest I don't know who "really" pays but I'm certain that "I don't see any fees on my bill and I paid the cash price" is not the right way to read the situation.
But as a consumer myself, leaving debt part of the equation aside (since it is irrelevant to the transaction itself), I pay zero transaction fees with my CC, whereas I'd have to pay a significant fee with a cryptocurrency.
More annoying is that the cryptocurrency fee would be charged every time I make a payment. Looking at my CC statement, I can see I & wife made about 50 CC transactions last month, which would result in a significant "monthly fee" if we had used Bitcoin for example (to pick the most "widely adopted" cryptocurrency).
Like I mentioned above, cryptocurrency has very specific key positive aspects, but low transaction fees aren't one of them.
Totally agree that on a practical basis it would be crazy to try and justify the utility of crypto to the average person based on "lower fees" since they are exorbitant for all but very large transactions.
My point with CC fees was, merchants "pay" but the open question is: are prices in a given market perhaps a bit (even just 1 or 2%) higher than they would be if card transactions were free? If so, exactly how much? Does this differ across different markets / product categories and how much? This is the kind of thing one would need to figure out to know who is "really" eating the fees.
My intuition is that whether the charges can be passed through to the consumer via higher prices is going to depend on the market (say supermarkets vs luxury goods) but I'm not aware of any research on this.