Copyright can start with an AI generated part
Assume that I tell an AI model to generate a specific sound for sampling. Then I take that sound and remix it to create an actual song from the sample. Now I have taken an uncopyrightable item and heavily transformed it.
I can register that song, because it is not just an AI product, but a human author's work based on an uncopyrightable sound. And the copyright would be valid, as there is human authorship in the remixing and making of the actual sample.
Licenses for access to property can exist for non-copyrightable items!
If I generate a song and do nothing to it before publishing, then it is still my property (depending on the AI model's license. Yes, there is no copyright. However, assuming I own it, I can legally sell you a license to access the file and have in the license terms that you can't share them.
Nobody is harmed by having to acquire a license to get access to an uncopyrightable item: if an item isn't available to be enjoyed anywhere but at a specific place, you have to pay the admissions fee to get there, no matter if the item in the place is under copyright or not. You can't see the real Mona Lisa anywhere but in the Louvre, you can't hear my AI-generated song anywhere but at these services where you might either get ads or pay for no-ads.
Note that I say the license is for access, not for the item itself. I don't have a copyright in the AI song, I only have a property interest.