I think it’s ultimately about whether training on copyrighted content is legal or not.
Here are some other quotes from the lawsuit that approach it from a different angle: “These tools also wrongly attribute false information to The Times.” “By providing Times content without The Times’s permission or authorization, Defendants’ tools undermine and damage The Times’s relationship with its readers and deprive The Times of subscription, licensing, advertising, and affiliate revenue.”
Even if the first argument fails, if the second argument wins, it still boils down to not being able to train on copyrighted content unless it is possible to train on copyrighted data without ultimately quoting that content or attributing anything to the author of that content. My (uneducated) guess is that’s not possible.