Copyright has always traditionally required there to be some sort of direct linkage to the source material, like "This has X character that I own in it" or "This is like X story I made, except Y and Z were changed".

Generative AI for the most part doesn't do that. There is no line to draw from their pictures to the AI's pictures. The lawsuit that maybe stabs these programs in the back would be a big artist claiming that they used the research LAION training set, knowingly, to create a product that copies their style exactly via their labeling of works with their name, and thus reducing their way to make money. Whether that has enough basis in law to work.... debatable.

But "This work it generated violated my copyright" is for sure not the way to get them.