Free speech hypocrisy?

As we all know, the First Amendment in the US Bill of Rights guarantees us five freedoms:

1 Freedom of religion
2 Freedom of the press
3 Right to assemble
4 Right to petition

aaaaaaaaaaaaaaaaaaaaaaaaaaand...

5 Freedom of speech

The First Amendment does not protect us from certain kinds of speech, including: libel, obscenity, threats, anything to incite violence or breach of the peace, etc.

This amendment is not to protect speech that is already accepted by society, but to allow offensive and otherwise unwanted things to be said.

However, this does not apply to private organizations. For example, Facebook is legally allowed to censor whatever you may want to post, whether it be bad language or something racist.

Even though these organizations have a right to do this...should they? This is something I have been thinking about lately, especially for libertarians (and other fierce protectors of the First Amendment).

Let's say I have rented out a small (10 x 10 foot) room in a building and had an array of shelves and tables for people to put newspapers, magazines, and other physical forms of media on. This is my "Alternative News" room - where people can learn more about current events other than from mainstream media outlets. Lately, I have noticed that Infowars newsletters have been showing up (and yes, I have heard that those actually exist), so I end up just throwing them out.



...Recycling them to be more precise (and environmentally-friendly).

Would doing this affect the integrity of my business (or whatever I have going on here) and make me a hypocrite?

Would Libertarian Facebook pages be hypocritical for promoting freedom, but censoring offensive speech?

This is a discussion post, so please participate.

EDIT: It seems that some people are misunderstanding what I am asking. As someone said on Facebook, I am not asking a legal question, but a moral one. I know that the First Amendment does not protect people from certain kinds of speech and that it does not protect employers, participants, etc. under a private organization from what they might want to do or say. What I am asking is if it is "right" for free speech advocates to censor what would otherwise be protected under the First Amendment if it was a public institution rather than a Facebook page or a private business.

All in all, it comes down to this: Is "it's a private organization - it can censor speech however which way it likes" - a legitimate excuse?"