Cloudflare isn’t a monopoly/chokepoint to the extent that actions against domain registries (via registrars) are.
What's more worrying is what happens when the publishers go back to court to complain that Sci-Hub are changing their name to avoid the block, from sci-hub -> sky-hab -> psy-hob -> buy-tub -> fly-tab -> ... At what point does the court just empower the publisher to shut down any site they choose, like a DMCA takedown request?
It's not that big of a problem as you think. even if there's no stipulation that it can only be used to shut down the science paper sharing site operated by Alexandra Elbakyan, it's the intent that matters. using it to shut down [unrelated site] will likely land them in hot waters with the judge.
I'm also not clear on the specific mechanics behind the injunction. Let's say that Ebalkyan does create sci-hub-sans-acs.org and claims all the copyrighted works are removed, can they still block it preemptively? Does Ebalkyan have to submit evidence that she's removed the copyrighted works? Can they demand action against anyone else that "seems" to be sci-hub? Is there a level of evidence they must meet? It kinda sucks to be ignorant of a legal system which can have so much impact in your life...