Speaking purely hypothetically, it's easy for a customer to carry both a Visa & MC, so if stores more frequently took only one or the other, customers would respond accordingly. Like they already do with AmEx, or when traveling internationally (where MC has a big leg up).
But that's chicken and egg. Collective action problem. You can't stop accepting one or the other until the large majority of customers carry both, which they won't do while most vendors continue to accept both thereby allowing the cardholder to only need one.
True, although Visa card circulation far eclipses MC, so you could take the gamble that they'll have a Visa. Last I checked, Visa actually had more cards in circulation than all three of the other major brands combined.
Slightly old info, but:
SEC filings for end of 2016 had Visa circulation at 335 million, MC at 200 million, Discover at 51.4 and Amex at 47.5.
The problem is it's not a gamble, it's a percentage. Even if 75% of your customers have a Visa, then 25% of them won't and you risk losing that amount of business by not taking what they do have.
Moreover, that still wouldn't give you competition, because then what incentive does Visa have to give you a good rate if they're the only network you could plausibly attempt exclusivity with? It's like trying to solve a lack of competition by creating a monopoly.
