Facebook removes all her groups. Cloudflare shuts down her websites. Youtube removes her videos. Google only leaves anti-Elizabeth Warren search results up, etc. Basically a major presidential candidate is completely locked out of having any pretense on the Internet whatsoever.
How many people who hold the above position, would say "yeah, this is fine. These are just private companies choosing who they want to associate with. Free speech doesn't apply here."
Congress shall make no law...
Although I suppose you could argue that once you have a handful of congressmen in your pocket, the First should apply to you too?
To answer your question more specifically: if Facebook is like a TV-channel, then they should be regulated like one: channels can broadcast what they want. If facebook is more like the TV medium then it's a different story. Then you can't have it arbitrarily cut access to certain interests. BUT in that case it's also probably "too big to have that responsibility in private hands" and something should perhaps be done about that (just like some banks may be "too big to fail" and as such can't be responsible for the economic damage they cause).
That implies having feasible alternatives when there are none.
Also, it implies you are aware information being prefiltered for you. Even if Warren knew about it and talked about it, she would sound like a conspiracy nut.
No alternatives to Google or Twitter? What?
Just because you don't like the venue or the service in no way makes them "infeasible". Just because you can't as effectively promote your cause doesn't make them "infeasible". There are other search engines besides Google, and there are other social media platforms besides Twitter, Facebook, and Reddit.
Any suggestion that a private company should be forced to broadcast speech that it finds distasteful is a direct affront to the 1st amendment, full stop.
8chan being denied access to another company's resources is in no way a free speech issue. In reality, 8chan's ideas are losing in the "marketplace of ideas" and now we have people who suggest government intervention to give their points of view a leg up.
Perhaps you could argue humans aren't built for internet communications and so free speech doesn't apply to the internet. But Supreme Court precedent disagrees. You could make a similar case that free speech doesn't apply to anonymous speech, which would have meant all the anonymous pamphleteers during the American Revolution would have had more difficulty propagating their (violent, revolutionary) ideas.
I don't think hosting is a problem for 8chan, though. They can probably find some hosting service that would in theory respect their free speech, in the absence of a court order. However, the second they go online again without cloudflare's protection, they will quickly get attacked and the hoster will have to shut it down.
Which means we now have, effectively, mob-enforced denial of free speech.
Cloudflare is the great equalizer. But when Matthew Prince wakes up on the wrong side of the bed and decides to nuke one out of many, many objectionable sites because he can't or doesn't want to take the political heat of proxying for some site where users talk trash or post manifestos about violent acts (which are hardly ever acted upon), cloudflare's protection can vanish. Then you have to shut up, because if you don't you're vulnerable to attack by the digital mob.
Twitter claims to be a public square, the WSJ does not.
It is censorship, even if it is private companies and nobody got arrested etc, censorship is surpressing undesired information
My point is I don’t think evaluating it on government/private business is very helpful in determining whether something should be allowed or not.
... yes? We would say that.
That's exactly how we draw the line now. For example, nobody would demand a stadium offer naming rights to the KKK, but the state of Georgia had to allow that organization to "adopt a highway" under its charity-for-recognition program (https://www.cnn.com/2016/07/05/us/georgia-kkk-adopt-a-highwa...).
Ultimately, there is no uncontested free speech right in private conduct. If I am exercising a speech right to (say) rent a billboard for an advertisement, then the owner of that billboard is also exercising one if they give me a discount because I'm advertising kittens for adoption.
Once the government gets involved, however, it is often acting with the force of law. That is where we deem free speech rights to be sancrosanct.
Once someone sets up their "private property" as a public forum, they're on the spectrum of operating as a de facto government. If there were ideological competition between reverse proxies like Cloudflare, the "private association" argument would have weight. But rather Cloudflare seems to be acting due to pressure from an concerted effort to implement censorship across every such business.
No one argues against the right of a company not to host Nazi forums. But there is an ongoing trend were companies that wished to be apolitical are forced into political decision.