The problem you're describing isn't so much that private companies are allowed to choose who they do business with, but rather that the US has allowed far too much consolidation of markets to occur.

Regarding your example: I fully agree that Visa and Mastercard have way too much control over payment infrastructure. Before doing anything else, I'd break up their de facto duopoly. I'd focus on Visa in particular, as they control nearly 2/3 the market by themselves.