Doesn't a business have the right to decide who they get to do business with? If a business sees a customer doing something they're not comfortable with, don't they have a right to refuse service?

They do, but once you are a certain size, decision you make, create much bigger waves. At that point, 'derisking'(as the banks call it ), can easily alter the economy in unpredictable ways. This is one of the reasons, you typically do not want monopoly/oligopoly in place. MC and VISA already have amazing amount of market power. If they say they won't process your payment, you only have few options left ( wikileaks donations debacle comes to mind ).

We need to distinguish between institutions so big that they are defacto system and institutions that are mere cogs of the system.

Your argument is a good one, but it ignores attributes of the parties.