Is there a case that actually says this? Why would whether something is fair use depend on that? For that matter, how would they even show that a given AI model was trained on something from a recursive crawler rather than the same articles added to the training data after being downloaded by hand?
Twitter griefs everyone with a login wall because they want bulk downloaders to pay for API access instead and the login wall is an attempt to rate limit non-API bulk requests.
That isn't relevant to ordinary media outlets because a) they don't have enough content volume for rate limiting to be effective since it's possible to get everything they publish even at a slow rate limit, and b) getting AI scrapers to subscribe to their bulk download API instead is not the objective in their case.