I absolutely do not see why a website owner, should be forced to host content that he considers reprehensible. I wouldn't want to be forced to host such content on here and I don't get why YouTube should not be held to the same level of standards.
The line of reasoning that it counts as a public space is odd, as it implies YouTube is a public service, sure if you want to nationalise such a service and then debate it, fair enough, but there is a basic concept of private property being private property at stake here. If you're not going to brake up the mega corps, then I don't see why we should treat them any differently to other things people own, such as homes.
I think the line of reasoning goes that this will slippery slope into actual government restrictions on speech. One issue is such an idea ignores the concept of psychological accounting on different types of speech, meaning there's going to be a rapid influx of emotions once a government did step over the line on what is and isn't considered reprehensible.
In turn, if we are to jump into the realms of simplistic political slogans like that, I think it would be reasonable to argue in turn that such concepts could also be legally twisted to justify forcing normal people to host content that in their mind is wrong. It would be very interesting to note that the whole gay wedding cake dramas are basically a variant on this argument, with the sides typically reversed.
Or we can agree that consent, power and slopes are far more complicated then that, which is why it's an informal fallacy for a reason.
I do not think restricting speech is always a certain block on extremism though, it certainly is harmful to have unfetted speech just flowing around and crowding out facts as the US shows; but I'd consider extremism more a result of the overall economic situation and banning Nazis is a plaster, but still something that doesn't address root causes, as the AFD in Germany shows.
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I wonder where the origin of defending the free speech of Nazis came from though, I have suspicions that it was started by actual Nazis themselves and dropped into a conservative audience, as a way of manipulating them. But now I'm just going into hypothesising here...
Because Nazis having been manipulating the right almost as long as they've been around, it's not hard for a malicious few to exploit the weakness of a majority, me thinks.