Does this matter in practice though? By modifying some of the generated code and not taking a solution produced by an LLM end-to-end but borrowing heavily from it, can't a human claim full ownership of the IP even though in reality the LLM did most of the relevant work?
I think as long as the human puts in substantial and transformational effort, they can claim to be the copyright holder of the entire work, yes.
Compare taking snapshots with a camera.
Because some photographer somewhere can claim to have put in a lot of effort, we all get IP protection for photographs by default.