Just a point of clarification: Copyright is about the right of distribution. So yes, a company can just “download the Internet”, store it, and do whatever TF they want with it as long as they don’t distribute it.
That the key: Distribution. That’s why no one gets sued for downloading. They only ever get sued for uploading. Furthermore, the damages (if found guilty) are based on the number of copies that get distributed. It’s because copyright law hasn’t been updated in decades and 99% of it predates computers (especially all the important case law).
What these lawsuits against OpenAI are claiming is that OpenAI is making a derivative work of the authors/owners works. Which is kinda what’s going on but also not really. Let’s say that someone asks ChatGPT to write a few paragraphs of something in the style of Stephen King… His “style” isn’t even cooyrightable so as long as it didn’t copy his works word-for-word is it even a derivative? No one knows. It’s never been litigated before.
My guess: No. It’s not going to count as a derivative work. Because it’s no different than a human reading all his books and performing the same, perfectly legal function.