If you pay your credit card bill off in full every month, you're not paying anything extra unless the merchant charges a premium for credit card transactions. If you carry a balance, you're paying on that, sure, but nothing requires you to do so.
(Also, of course, you might be actually getting 1% or more back on credit card transactions, if they have cash back programs.)
The merchant is paying extra, and that cost is hidden in the price I pay.
If what you really meant is that the cost of the credit card transaction is implicitly in the price everyone is being charged, then sure, I see your point -- except that I think it actually strengthens the original poster's argument for using credit cards! If you pay $25 for a widget in cash and I pay $25 for the same widget on my credit card, then we're literally both paying the same price, but I'm getting the ostensible benefit of using the credit card and you're not. (I would actually argue, unlike the OP, that this is mostly true for debit cards as well, in that you still have a bank fighting to get your money back in a way that isn't true with cash.)
We also have an "EFTPOS" network that is independent of Visa/MC. It's a debit account network owned by the banks that give you access to savings or cheque accounts. It's charges are much lower than Visa/MC, so merchants like when you use it and don't apply a surcharge.
Even cash likely has a higher fee, if not from simple counting mistakes on many small transactions, then complexities like the safe, the risk to employees (insurance), the controller effectively managing the cash each day, the security of a bank truck, and more.
For me, it's always been about that buffer between my actual money and the merchant. Sure, credit card rewards programs have been great, but having a middleman who is more or less on my side in all financial transactions is a game changer.
If I use my debit card, the money is out of my possession; it's possible to get it back, but it's not certain, and while I'm waiting for that to happen, I don't have access to that money. (And if I use cash, I pretty much just have to accept that the money is gone the second it leaves my hands.)
If I use a credit card, I still have the money, and can dispute the charge. Even if I don't notice the problem until after I pay my credit card bill, I still have a single entity (who, again, is more or less on my side) who will refund my money if I have a valid complaint, even if the merchant isn't playing ball or is a fraudster.