I've studied copyright for over 20 years as an amateur, and I used to very much think this way.
And then I started reading court decisions about copyright, and suddenly it became extremely clear that it's a very nuanced discussion about whether or not the document can be copied without explicit permission. There are tons of cases where it's perfectly permissible, even if the copyright holder demands that you request permission.
I've covered this in other posts on Hacker News, but it is still my belief that we will ultimately find AI training to be fair use because it does not materially impact the market for the original work. Perhaps someone could bring a case that makes the case that it does, but courts have yet to see a claim that asserts this in a convincing way based on my reading of the cases over the past couple of years.