My (albeit limited) understanding is that Visa and Mastercard ensure that the debit/credit transactions are securely and quickly occur, but if there are post-transaction issues like fraud or chargeback, then the credit card-issuing bank handles/resolves those issues.
Conversely there's probably a subset of transactions that don't need this sort of "insurance", such as buying groceries. Currently the merchant still has to pay the 2%+ fee in the U.S. but would be nice to have the option for a customer to waive the "insurance" part and benefit from a 2% savings. It's akin to many merchants offering a lower "cash-only" rate.