It seems like you're working under the core assumption that the trained model itself, rather than just the products thereof, cannot be infringing?

Generally if someone else wants to do something with your copyrighted work – for example your newspaper article – they need a license to do so. This isn't only the case for direct distribution, it includes things like the creation of electronic copies (which must have been made during training), adaptations, and derivative works. NYT did not grant OpenAI a license to adapt their articles into a training dataset for their models. To use a copyrighted work without a license, you need to be using it under fair use. That's why it's relevant: is it fair use to make electronic copies of a copyrighted work and adapt them into a training dataset for a LLM?

You also seem to be assuming that a generative AI model training on a dataset is legally the same as a human learning from those same works. If that's the case then the answer to my question in the last paragraph is definitely, "yes," since a human reading the newspaper and learning from it is something that, as you say, "any intelligent rational human being" would agree is fine. However, as far as I know there's not been any kind of ruling to support the idea that those things are legally equivalent at this point.

Now, if you'd like to start citing code or case law go ahead, I'm happy to be wrong. Who knows, this is the internet, maybe you're actually a lawyer specializing in copyright law and you'll point out some fundamental detail of one of these laws that makes my whole comment seem silly (and if so I'd honestly love to read it). I'm not trying to claim that NYT is definitely going to win or anything. My argument is just that this is not especially cut-and-dried, at least from the perspective of a non-expert.