>big corporation that are monopolies or oligopolies

Monopolies and Duopolies suck even with regulation. If you want a state owned payment processor, pursue that goal. But you will end up in the same place, with reactionary leaders using the service to ban things that the voters disagree with, which sometimes will be porn in all likelihood.

>"private companies" as if that means something.

A private company can be competed with. You can legally pay these websites just not via those 2 most popular options. If something is actually censored, the state will use violence to prevent you accessing it. Theres a video kicking around the internet of NSW cops snatching a VHS tape from the hands of people trying to run a small banned film festival. Thats definitionally censorship. I know you want to take the anger people have at censorship, and transfer it away from the government to people you dont like, but its simply not censorship.

> Due to their power, their decisions affect people as if they are government decisions

The inertia of people who wont change their purchasing decisions to avoid those to options is the issue. Not their choice not to process payments.