I get the feeling, but that's not what this is.
NYTimes has produced credible evidence that OpenAI is simply stealing and republishing their content. The question they have to answer is "to what extent has this happened?"
That's a question they fundamentally cannot answer without these chat logs.
That's what discovery, especially in a copyright case, is about.
Think about it this way. Let's say this were a book store selling illegal copies of books. A very reasonable discovery request would be "Show me your sales logs". The whole log needs to be produced otherwise you can't really trust that this is the real log.
That's what NYTimes lawyers are after. They want the chat logs so they can do their own searches to find NYTimes text within the responses. They can't know how often that's happened and OpenAI has an obvious incentive to simply say "Oh that never happened".
And the reason this evidence is relevant is it will directly feed into how much money NYT and OpenAI will ultimately settle for. If this never happens then the amount will be low. If it happens a lot the amount will be high. And if it goes to trial it will be used in the damages portion assuming NYT wins.
The user has no right to privacy. The same as how any internet service can be (and have been) compelled to produce private messages.