Rereading this it sounds like you're defining "person" as "capable of generating usable training output for LLMs."
Even if LLMs do become capable of generating usable training output for themselves, they will still not have human personhood.
Personhood as a moral or legal or consciousness definition, sure.
Personhood as a capacity to participate an an agent in a network of mutual recognition of personhood, however, is likely.
https://meltingasphalt.com/personhood-a-game-for-two-or-more...