You seem to misunderstand what an LLM does. It doesn't generate "right" text. It generates "probable" text. There's no right or wrong since it only generates a single word ahead of where it currently is. Hence why it can generate information that's complete bullshit. I don't know the details about this Go AI you're talking about, but it's pretty safe to say it's not an LLM or uses a similar technique to it as Go is a game and not a creative work. There are many techniques for creating algorithms that fall under the "AI" umbrella.

Your second point is a whole different topic. I was referring to a "derivative work", which is not the same as "fair use". Derivative works are quite literally everywhere. https://en.wikipedia.org/wiki/Derivative_work A derivative work doesn't require fair use, as it no longer falls under the same copyright as the original. While fair use is an exception under which copyrightable work can be used without infringing.

And also, those projects most of the time do not get shut down because they are actually illegal, but they get shut down because companies with tons of money can send threatening letters all day and have a team of high quality lawyers to send them. A cease and desist isn't a legal enforcement from a judge, it's a "recommendation for us not to (attempt to) sue you". And that works on most small projects. It very very rarely goes to court over these things. And sometimes it's because it's totally warranted. Especially for fan projects it's extremely hard to completely erase all protected copyrightable work, since they are specifically made to at least imitate or expand upon what they're a fan project of.

EDIT: Minor clarification