We seem to agree that "lower prices" have no place to come from. If "lower prices" can't happen, what are the benefits to users now?
Alternatively, merchants can give users more cashback points for using their preferred way of paying. Not sure how this flies with payment processor rules, but I've seen it in the wild.
In this case, people who don't have local cheaper option won't be able to buy from them.
Cash/local-card-only might be OK for a grocery store in rural area only serving the locals, less so for a coffee shop in tourist centre of a major city.