Sure but who is at fault?
If I manually type an entire New York Times article into this comment box, and Lemmy distributes it all over the internet... that's clearly a breach of copyright. But are the developers of the open source Lemmy Software liable for that breach? Of course not. I would be liable.
Obviously Lemmy should (and does) take reasonable steps (such as defederation) to help manage illegal use... but that's the extent of their liability.
All NYT needed to do was show OpenAI how they go the AI to output that content, and I'd expect OpenAI to proactively find a solution. I don't think the courts will look kindly on NYT's refusal to collaborate and find some way to resolve this without a lawsuit. A friend of mine tried to settle a case once, but the other side refused and it went all the way to court. The court found that my friend had been in the wrong (as he freely admitted all along) but also made them pay my friend compensation for legal costs (including just time spent gathering evidence). In the end, my friend got the outcome he was hoping for and the guy who "won" the lawsuit lost close to a million dollars.