The verbatim reproduction of their copyright is the interesting point. I can see where it would be free use to train on content but not to reproduce it.

I feel like it is clearly a copyright issue if someone, through whatever means, posted New York Times articles on the internet for 1000s to potentially see and find through search.

If someone through some other interface provides 1000s of people independently through private channels a New York Times article verbatim on ask isn’t this similar?

In both cases it requires users to seek it out. In both cases copyright protected content is openly available to 1000s of people. Just because you generate your stock of copied material 1 by 1 on an order by order basis doesn’t feel materially different in effect on the copyright owner. The interface owners are willing to distribute copyright protected content 1000s of times on request for it.