Imagine if a paper newspaper were told that they had to publish every letter sent to them or no letters at all. They would stop publishing letters entirely. There are a lot of ways to construct laws such that they have the opposite of their nominal effect. It's important to note that the US First Amendment bars the government from passing laws respecting both freedom of speech and freedom of the press (among other related rights).
Wrong analogy. Newspaper does not offer a service of publishing customer letters. They offer a service of receiving and reading letters on premise that they may choose the letter to be published. It's a different story.
If a newspaper published all letters, then government could act on behalf of citizen to protect freedom of speech, by making sure there's no arbitrary censorship of publications.
The relevant detail is that you can indeed censor a publication by adding a rule that says "you can't censor your publication in the following ways". I don't personally use Facebook at least in part because I don't want to use a centralized, censored system, but the fact of the matter is that it's a bad idea (not to mention likely an unconstitutional one) to tell Facebook what they cannot publish and indeed what they must publish.
Although it's irrelevant to the point, I also don't totally understand the nature of your objection. If you are saying that any publication gets to define what it does and does not publish (i.e. Facebook says "I am a company that publishes non-nude photos" or "I am a company that publishes censored material"), then they merely have to re-define their service in order to censor people while complying with whatever scheme you have in mind.
Facebook is not a newspaper, because it does not produce the content that it makes available - freedom of speech cannot be applied to Facebook itself. FoS applies however to the content producers and limits the terms of contract that Fb may offer to them, by banning the arbitrary censorship (the distinction between artwork and offensive content is arbitrary and decided by Fb moderators). Also, by saying "unconstitutional" which constitution do you apply to it? It's multinational and has the business in many countries - each one can regulate Fb within its sovereignty.
