If the work goes beyond fair use, it is a copyright violation. It doesn't matter if it was created by a person or an AI.

Technology that makes copyright violations easier/quicker have typically been found legal if "the technology in question had significant non-infringing uses".

This makes sense. It was allowed for the content to be read and used in certain ways (e.g. search engines or as references) without substantial reproduction. The NYT would then have to flag specific generated content as infringing a specific work which could then be judged as fair use or not on a case-by-case basis. If a particular site/company was repeatedly and/or primarily using substantial content then perhaps it could be 'delisted' as search engines do for links to pirated copies of works.

It really hinges on substantially similar. If I copy Harry Potter and change every instance of Harry Potter to Michael Rose surely it's infringing. If I write a coming of age story set in a magical land I'm probably OK. Which do you think LLM produce?

Whatever you ask and the model will be judged by its accuracy.

If you ask for Harry Potter and it gives you Bart Simpson it’s useless.

It's likely not possible of literally giving you Harry Potter. If you specify it narrowly enough that it qualifies as fan fic its probably exactly what you were going for. After all your word processor is capable of producing infringing works but is not itself an infringing work.