You are related when you try to access a site. It's just customer service issue. You can't demand that sites allow you in.

   you <---> C <---> site


   you <--X--> C <---> site
          |
         Court order

See the difference.

In the second case, the parties are still related. The websites that are intended to be targeted by the court order are served by Cloudflare, and the operators of the sites that you want to access are also served by Cloudflare. It is like doing business with a bank that also serves sanctioned customers, and now your suppliers cannot get paid.

Can Cloudflare demand that ISPs carry its traffic? Probably, due to net neutrality laws. That's what they are trying to do in court.

Can you demand that websites allow you in? Depends on the site, I can imagine certain kinds of sites, e.g., government websites or public utility websites, being compelled to do this by a court, if they use Cloudflare and block innocent users. But the blocked users will generally not have enough time or money to deal with a lawsuit.