I think if a person writes something and just uses AI to clean it up or tweak it, it’s still their work and should be protected by copyright. But if it’s entirely generated by AI then probably not. That said, in the United Kingdom (at least from what I understand) computer-generated works can still be protected even without a traditional human author. In that case the author is basically whoever made the arrangements for the work to be created. But honestly, I feel like this is likely to change. The whole area is still pretty unclear and hasn’t really been nailed down yet.

Suno (an AI music site) specifies in their FAQ that the music isn't copyrightable. But your lyrics might be. And if you put a project together (and album or whatever) then that might also be. If you orchestrate a video for it, that is likely copyrightable. But the music itself isn't, since it was an algorithm based on a lot of other people's work that made that happen - not your skill or effort.

People who use AI to improve their own work most definitely have an argument that they own a copyright to the output. But in this case, they might only have claim to the lyrics, for example.

In my opinion, AI generated output should be copyrightable as long as, and only if, the person using AI prompted it to generate the output based on an idea they originally had. If the idea itself was AI generated, I don't think it deserves copyright. But if the idea was original and AI was just used as a system to do the heavy lifting (coding etc.), then I don't see an issue with being able to copyright the output. This is just my opinion.

Depends on where you are (copyright law varies by country!). In the US courts have ruled that computers can not be creative so anything generated by a computer is by default not copyrightable. If there is enough creativity included into the work by a human operator, then the work does become copyrightable.

And for should it be, I think this is a fine situation. Don't see a need to change it.