The Courts have broad leeway over document retention in a legal proceeding. The fact the documents are bring retained doesn't immediately imply plaintiffs get to see all of them.

There are myriad ways courts balance privacy and legal-interest concerns.

(The Times et al are alleging that OpenAI is aiding copyright violation by letting people get the text of news stories from the AI).

If people can get the text itself from AI, then anyone can then why would it need access to other people's data?

Does the Times believe that other people can get this text while it can't get it itself? To prove that the AI is stealing the info, Times does not need access to people's logs. All it has to show is that it can get that text.

This sounds like Citizen United again to AstroTurf and gets access to logs with a fake cause.

It's not whether people can get the data. They need to prove people are getting the data.

So in effect if a lot of people don't get the data now, then it will never matter, is that right?

That logic makes no sense because if they don't get it right now then it does not mean that they will not get it in future.

If Times and its staff can get the text, is all that matters because the use and rate of data usage is not material as it can change any time in future.

Court cases aren't generally about hypothetical futures. There is a specific claim of harm and the plaintiff has a legal right to the evidence needed to prove the harm if there's reasonable suspicion it exists.

Capone isn't allowed to burn his protection racket documents claiming he's protecting the privacy of the business owners who paid protection money. The Court can take steps to protect their privacy (including swearing the plaintiff to secrecy on information learned immaterial to the case, or pre-filtering the raw data via a party trusted by the Court).