I think the characterization is unfair to generative AI, because the information that is retained in the model is a very lossy and rough generalization of the training data - that situation is a lot less direct than what it's like with search engines. In my mind, the lawsuits against search engines have a lot more ground to stand on. Saying that something "provides access to copyrighted content" almost implies that there's some mechanism that allows the user to wholesale download complete, unedited and exact copies of some copyrighted material - I could see that argument with a search engine, but not really with a text or image generator.
It's 100% dependent on copyrighted material at least, even though you can't access an exact copy of it without access to the material it wouldn't exist. It's a messy issue, I have no idea how it will be solved since it's a rather new development for humanity, in the past when humans would collect knowledge from different sources to use them in a novel way there would be at least some kind of attribution, recognition of the sources, either being cited or acknowledged in a preface. With GenAI there's nothing, and probably not even a way for GenAI to tell us where it got "inspired" from to generate something.
It's going to be a very messy landscape for copyrights and intellectual property in the next years.