Matter of scale, matter of society
It is true that the First Amendment only applies to government. That is because it is part of the constitution, not because that is the limit of the intent of the ratifiers. The ratifiers were committed to the principle of Free Speech _in society_, not just relative to government. They knew that the government is a natural vector of oppression, so when they formed this one, they did so on the condition that a Bill of Rights be added. It is not an accident that the amendment first adopted was this one.
But government is not, by far, the only instrument that people can use of oppression. In fact, we regularly use the government to shut down oppressive behaviors by other entities. Even the government/non-government distinction is not that clear. We allow states to vary their laws, not just to test them before national application, but because people can (and do) move between states in part because the agree or disagree with the policy of government in those states. National media attention is often brought to bear on HOA restrictions for things like flags. In fact, the HOAs win these fights in the courts. They sometimes loose them in the court of public opinion.
If I have a comic strip, blog, or newsite, even one with a global following, whatever constraints that I want to demand of commenters is my business. Why? Because there are literally tens of thousands of other equivalent places people can go to express their views, and I will lose ad revenue, or at least my own voice, if they do.
But...what happens when there is a pervasive exclusion based on weak criteria? The Jim Crow laws were overturned, so that governments could not discriminate based on race, but how much difference did that make? In fact, discrimination by businesses and associations became more intense as people holding these views resented the repeals. Privatization also became a popular dodge.
Marxists theory explicitly calls for this sort of exclusionary behavior. Why should the political views of a professor of mathematics affect their job prospects? Of someone in IT? And yet expression a libertarian or conservative view will get you targeted in much of our industry.
Ever see a comedy bit where the commander asks for a volunteer to step forward & everyone except one mug takes a step backwards? That is what we are seeing as it regards free speech on the internet. The internet allows "anyone" to have a platform to express their views to an extent undrempt of by any but a few just fifty years ago. As long as you don't offend (current) liberal sensitivities. Then, you will hounded out of every forum which they control--whether you expressed said opinions in a particular forum or not.
THIS is the complaint against Monroe's comic. The "more equal" brigade is on the march, driving unhygienic views out of the public sphere. The marketplace of ideas is becoming decidedly unfree. And he defends this.
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As for the particulars of this case, if they were calling for violence, then they were crossing the line. They need to be shut down along with Antifa & BLM. Oh, wait. Antifa & BLM are NOT being shut down, you say? Why exactly is that?
Was Der Stormer calling for violence? I've never heard that they were. And yet, they had to go to China to find a registrar. If you are completely confident that your views will remain acceptable to the constantly evolving leftist intersectionality, then good for you, I guess. I, for one, don't welcome my new overlords.