Nobody "owns" it for the most part. Most LLM output can not be copyrighted, because there is not enough human essence in it. Meaning that it is public domain the moment it is produced.
There might be a few exceptions, if you really guided your model in a certain particular way, but that is rare.
What /could/ be is that you violate their ToS. If that ToS is valid and legally and practically enforceable is a different question and will depend a lot on your jurisdiction.
> Subject to your compliance with our Terms, we assign to you all of our right, title, and interest—if any—in Outputs.
Originally it was not that way. They changed it because, as you suggested, their service has questionable utility without it.
The outstanding one is the “meaningful human input” part of copyright. The most recent ruling is that prompting alone does not count. If you write/rewrite sections, those are yours. Everything in between is a somewhat untested.