The right of free speech is not wholly encompassed by the First Amendment to the United States Constitution.

In fact, it's the other way around, it's because the right of free speech is recognized as a universal, natural right then the US Federal Government is not permitted to make a law suppressing speech. The First Amendment does not create the right. The right is there, naturally, whether or not the United States or its constitution or government exists. The First Amendment merely explicitly states that the government isn't permitted to impede that right.

Using the existence of the First Amendment to narrow free speech as a right to what the government is permitted to do and nothing else is a severe perversion of both the document and the beliefs of the framers.

In short, "it's a private entity doing it" is an incredibly poor defense of behavior that suppresses speech. It's like how young children will defend their rude or offensive behavior with "it's not illegal." The reason that's an unconvincing argument is that it's an incredibly low bar. The world is full of behaviors that may not be so universally offensive or outrageous that people have explicitly written down that nobody is every allowed to do that thing. It's actually a very small range of possible behaviors that that covers.

The only reason that there isn't a general law barring private parties from restricting the speech of others is (a) one's right to free speech does not necessarily negate another's rights in the same or a different area, (b) one's rights do not entitle one to the use of things owned by others against their desires, and (c) any such law could be used by the government to indirectly suppress other rights.

The narrow nature of the First Amendment is not to be taken as an implication that the right is narrow. It's an admission that the law cannot perfectly protect human rights.