Nobody, not even Anthropic, is arguing that they should be able to host other people's paid content for free. The crux of their fair-use defense is that models are transformative works, just like parodies or book reviews, and hence should be treated as fair use.
You can't just take a pile of books (no pun intended) and turn that into Claude in a day with 30 lines of Python, there's a lot of work and know-how on the Anthropic side that goes into making a good LLM.