These types of lawsuits probably won't work in the EU because the EU has already exempted training from copyright restrictions. It's already a settled question over there.
Even European authors will be able to sue a system that produces unauthorized reproductions of their work. Just because you can train a model, you still don't necessarily have the right to sell or give away access to it (unauthorized distribution and reproduction of copyrighted work), and will probably be liable if you publish any of its output that is plagiarism of copyrighted work. It will be interesting to see if the AI companies succeed in pushing the liability onto the end users when it hits the courts. And if industries adopt techniques like 'trap streets' to catch violations.