Does copyright protect the right to make copies

Among other things, yes. I think this is what this particilar lawsuit is about.

It will be really interesting to see whether they define an LLM as singular or plural/corporate. Those files (with hundreds of texts) seem to have been doing the rounds so it doesn't look like a single use to me. But I can also see the merit in your one AI = one student argument.

Re: your Agatha Christie example, not sure how that works in the US but in my country (New Zealand) if a book is in a library, then the author or publisher gets a certain yearly compensation payment based on how many copies are in how many libraries.

E-book licensing similarly has different costs based on how many "copies" ir simultaneous sessions a library is authorised to have.