You're correct, but no one has had the balls (or the lawyers) to clarify this in court yet. Expect to see hosting providers complying with takedown requests for the foreseeable future.
I don't remember the details (or outcome) but there was a lawsuit a few years ago involving CAD or architecture software and whether they could limit how the output images were used because they were assemblages of clipart that the company asserted were still protected by copyright. Something like that. A lot of "AI" output potentially poses a similar issue, just at a far more granular level.