I think -- could be wrong -- the worst a site a like this would suffer in that way are
Denial of Service (DoS) attacks, assuming the sites Admins and Owners take the most basic of precautions. (e.g. not using "password" as a password, that sort of thing.) The sort of 1990s movie silliness of HaX0r kings using their mad 1337 skillz to take down the Pentagon or CIA or whatnot have always pretty much been the realm of fiction. Probably the most famous DoS attack is the one simultaneously carried out
against Playstation and XBox on Christmas day 2014. The people behind that, to their credit, did catch the SysAdmins with their pants down, but everything was cleared up in under 24 hours. (I remember it fondly because I was trying to hook up a brand spanking new XBox One for a relative's kid, and for the life of me had no idea what I was doing wrong.)
I'm not sure what would happen if Cloudflare refused to support the site, but I do know there are sites they drop support for and this does somehow make a site far more vulnerable to DoS attacks discussed above.
Anyways, I'd think a much, much, much bigger issue would be with the webhosts refusing to allow the site's content. Simply because something is legal in the USA doesn't mean webhosts are required to support your content, after all. And most don't want any controversies. I can easily see a group like #FixThe26 pressuring webhosts to drop this site or similar.
At some point I'm guessing we'll start seeing actions against VPNs, possibly via something like China's "social credit" system effectively ending an anonymous internet. Nanny states like New Zealand and Germany are almost there as it is. And not just big government, big tech as well. I swear I read somewhere that the latest updates to the MacOS can basically override VPN functionality, but I can't lay my hands on a link. (Maybe it was their next generation of chips?) And we shall see what goodies are lurking in Windows 11, once that rolls out.
tl;dr - plenty of reason to be worried, just not IMO from the direction OP was thinking.