I've heard the "Private corporation, doesn't have to free speech" argument a lot lately, and I think it's pretty drivel, because that's not really what free speech is. Free Speech doesn't only extend to protection from government censorship in every way, it extends past that. Legally, sure, that's where the line is. The US constitution, and indeed most countries' laws or constitutions, only protects it to the extent of what the government can or can't do.
However, the conversation we're having shouldn't be one of simple legality. I mean, one, that's a terrible conversation, yes, the media platforms can legally do this, end of story.
While this definitely isn't illegal, it's a very bad principle for people to try and defend. Freedom of speech as a concept is based around the simple idea that ideas should flow and be discussed, and we should try to protect the freedom of unpopular ideas to be expressed, because without that, we can't actually improve our ideas and see which ones are correct and so on and so forth. In this regard, Alex Jones getting kicked off these media platforms are a really serious thing, and something we should not support.
While it's again definitely legal, having people support these massive and critical tools of thought exchange to have freedom of speech stricken from it only leads to bad precedence and the erosion of freedom of speech, and that's pretty fucking awful for all of us.