Replace the AI model by a human, and it should become pretty clear what is allowed and what isn’t, in terms of published output. The issue is that an AI model is like a human that you can force to produce copyright-infringing output, or at least where you have little control over whether the output is copyright-infringing or not.
>>layer8+(OP)
Its less clear than you think, and comes down more on how OpenAI is commercially benefiting and competiting with NYT than what they actually did. (See four factors of fair use)