*: Not always the same as Sam Alito's
About the OP, government is the right place where we "fight it out" and try to sloppily design a system to move forward as one; these seemingly activist campaigns to "jawbone" private companies is absolutely a sign that something needs to be done at the government side. However, there are very entitled, rich interests behind the banking system, so yeah, there isn't one easy solution i think.
Likewise, if a casino or betting company (ladbrokes, for example) have customers that win too often, I also think it should be illegal to stop them betting. Fundamentally if you're running a business that is an uneven coin toss (to your favour) and you have customers that are able to make money off you - that's your fault for having a bad business model.
So to answer your question, any size.
If you're a consultant do you have no right to refuse a client? Even if you have other clients you'd rather work for, or that particular client is a bad fit for you, or any other reason?
If you run a transport company, and you think someone is trying to get you to move illegal goods, or goods that you have moral qualms about transporting (such as a vegan being asked to transport livestock for slaughter) do you have no right to refuse?
Consultant - unless you've got a legitimate need to reject providing services to them, I tend to think the same, you should have to serve them if they're trying to pay you, or there's a legitimate business need to avoid that client.
Transport company - it's not your job to judge what's being moved. It's your job to move something from A to B. If you want to avoid moving livestock, don't go into the transport business. Should that same vegan be allowed to not teach kids in school because the kids they teach eat ham sandwiches? Should they be allowed to reject someone from banking services just because they own a fur coat?
But not wanting to concrete that person's driveway because they're in the army, or because they're a politician or whatever else along those lines I think shouldn't be allowed.
Large entrenched companies have leverage small businesses do not, in the same way that a large moon orbits in a way a test particle of infinitesimal mass does not. We already recognize this with respect to monopoly law: you lose your right to do certain things to your competitors precisely when you're large enough that you could reasonably suppress them.
That is essentially what we are talking about here: a duopoly that is actively suppressing competition. My understanding is that the big-two payment processors don't just refuse to process certain payments, they also refuse to work with banks who work with payment processors who will. Assuming that I am correct in that understanding (I might not be, this is not my area of expertise), that would prevent (or at least hinders) someone from just saying "there is a market need here" and forming their own payment processor to fill that need. To me, that seems like a problem for the exact same reasons that monopolies are a problem, and regulating against monopolies is not particularly controversial.
