Consider how commercial search engines are fine to show text snippets, thumbnails and site caches.

AI developers will most likely rely on a Fair Use defense. I think this has a reasonable chance of success since, while the use of a given copyrighted work may affect the market for that work (in this case NYT's article), it can be argued to be highly transformative usage. As in Campbell v. Acuff-Rose Music: "The more transformative the new work, the less will be the significance of other factors", defined as "whether the new work merely 'supersede[s] the objects' of the original creation [...] or instead adds something new".

There's also potential for an "implied license", as in Field v. Google Inc for rehosting a snapshot of a site, where "Google reasonably interpreted absence of meta-tags as permission to present 'Cached' links to the pages of Field's site". As far as I can tell in this case, NYT's robots.txt of the time was obeyed, which permitted automated processing of all but one specific article for some reason.