Isn't it, though?

I see people are still desperately clinging to this "websites aren't public services" narrative but as the years go by it's becoming increasingly clear that it just isn't entirely true.

The whole logic behind the "it's a private space/service so they can deny service to whoever they want" argument is that, if a user is denied service, they can seek service from an alternative provider elsewhere, usually without much difficulty. If the other providers also don't want to provide the service, fair enough. But this isn't really the case in the modern web because you often don't have alternatives. If you are denied service by cloudflare, you can't just go to another cloudflare down the street. Same with many other major online services that are effectively monopolies in their field.