Although the only thing actually protected by the First Amendment is protection against censorship of speech by the government. Private commercial enterprises can do what they want (since technically you should be able to move to a competitor).

1. Even if the First Amendment is only limited to the government, the concept of free speech predates and extends far beyond the First Amendment.

2. Does the Constitution apply if the government allows private entities to take over roles of the government? There is the concept of utilities and common carriers that extend upon this logic.

3. Even without the above points, one should still be able to see a double standard being applied to violent/extremist speech. Saying a company can do whatever it wants doesn't explain what the company actually wants. If the company is applying a double standard to what speech it allows, then the true wants of the company is something we need to discuss and decide do we want to tolerate.

An interesting example for your comment might be the use of private prisons, and probation companies.

This is also dangerous for clear reasons. If speech deemed bad is censored we may have no examples with which to contrast good ideas. Think of free speech as natural selection for ideas.

John Stuart Mill thought free speech should extend into the private world:

Society can and does execute its own mandates: and if it issues wrong mandates instead of right, or any mandates at all in things with which it ought not to meddle, it practises a social tyranny more formidable than many kinds of political oppression, since, though not usually upheld by such extreme penalties, it leaves fewer means of escape, penetrating much more deeply into the details of life, and enslaving the soul itself. Protection, therefore, against the tyranny of the magistrate is not enough: there needs protection also against the tyranny of the prevailing opinion and feeling; against the tendency of society to impose, by other means than civil penalties, its own ideas and practices as rules of conduct on those who dissent from them.

This is precisely true. The Constitution does not offer protection from the vagaries of capital, but capital realizes that certain ideologies, like brands, are toxic. Thus the New Yorker and Economist covers this week depicting the President' relationship to extremism.

Why not? If I own a bar I shouldn't be allowed to decide what entertainment I hire based on their content? We can't have a society where all speech is protected, even from private actors.