I think you are oversimplifying this issue and ignoring the context and purpose of using their content. Original analysis of data is not illegal, and that's all these models are, a collection of observations in relation to each other. As long as you can prove that your storage was noncommercial, and no more than necessary to achieve your fair use objectives, you can get by.

Fair use protects reverse engineering, indexing for search engines, and other forms of analysis that create new knowledge about works or bodies of works. Moreover, Fair use is a flexible and context-specific doctrine, and you don't have to prove in court that you comply with every single pillar of fair use. It depends on the situation and four things: why, what, how much, and how it affects the work. No one thing is more important than the others, and it is possible to have a fair use defense even if you do not meet all the criteria of fair use.

You're right about copyright forbidding much more than people think, but it also allows much more than people think. Fair use is also not a weak or unreliable defense, but a vital one that protects creativity, innovation, and freedom of expression. It's not something that you have to prove in court, but something you assert as a right.