You keep making the claim that because it was scraped people can do whatever they want, as scraping is legal. That is the only thing I'm arguing against, because that is a gross misinterpretation of how the case that made scraping legal was decided. LLMs aren't relevant to that point (which is exactly what I keep saying- the method of collection doesn't magically change the legality of it).
That being said, you're still wrong. The USPO has said that the output of LLMs are the outputs of algorithms and are not creative works. Therefore you can't "own the copyright to the new work you make" because the work itself can't be copyrighted at all. No one can own the output of an LLM.
Also, just because it seems you want to be wrong on every level, it is absolutely possible that a neural network would be able to repeat data from its training set. This is an incredibly known problem in the field.
https://www.bloomberglaw.com/external/document/XDDQ1PNK00000...
To put it another way, it's legal for me to go to the library and borrow a DVD or a book or poems. That doesn't give me the right to publish the poems again under my own name. Whether I find the poems from scraping, borrowing the book from a library, or even just reading it off of a wall I don't get ownership rights to that data.
The same logic applies to a lot of other laws around data. If you collect data on individuals there are a bunch of laws that come up around it, and many of them don't really concern themselves with how you got the data so much as how you use it. The fact that it was scraped doesn't grant any special legal rights.