Earlier quoted context omitted.
If Steam dropped Visa and Mastercard and successfully got most of their customers to use a Bitcoin payment processor instead, wouldn't these organizations just pressure that processor? I don't see how it would be any different. If the problem you're trying to solve is that private payment processors can deny service to anyone they don't like, the underlying technology the processor uses is irrelevant.
Bitcoin does not require any payment processor. That's its whole point: censorship-resistant, permissionless, p2p payment system.
Even companies working in large amounts of cash end up hiring companies to handle the cash logistics and all the other complications in dealing with it.