If you don't have a license to download the material, aren't you doing piracy?

If the licenses say that download is only permitted if you agree not to train on it or similar, then if you have downloaded it, you have agreed to the license, have you not?

I don't think it works like that. I can't just write "By reading this comment, you have agreed to pay me $100." and expect that to be legally enforceable unless you explicitly agreed to a contract to that effect in advance before reading the comment.

I could write "If you don't agree to pay me $100, I reserve the right to block your future access to my comments." (Which is what a ToS is.) But that's not a license agreement.

There's no such thing as a license to download. Downloads are requested, it's the sending server that's responsible for making the copy.