You're confusing the concept of free speech with the First Amendment. Any time a person is prevented from expressing themselves is a violation of their freedom of speech, even if they have no legal right to speak.

But even in the context of the First Amendment, freedom of speech does not only apply to the government. For example, net neutrality laws prevent ISPs, which are generally private companies, from restricting Internet traffic on free speech grounds.

To the extent that it is legal for a payment processor to censor speech, the only reasonable conclusion is that the law is wrong and must be amended. Large corporations are much more similar to governments than they are to private so individuals, and should be treated as such.