In the context of Shakespeare, I'd agree that there may be some competitive potential in the product. But in the context of news, something that evolves and relies on timely and accurate information, I don't see how something like that turns into competition for the NYT by being trained on past NYT outputs.
If the argument is that people can use ChatGPT to get old NYT content for free, that can be illustrated simply enough, but as another commenter pointed out, it doesn't really seem to be that simple.