When you use (read, view, listen to…) copyrighted material you’re subject to the licensing rules, no matter if it’s free (as in beer) or not.

You’ve got that backwards. Copyright protects the owner’s right to distribution. Reading, viewing, listening to a work is never copyright infringement.

This means that quoting more than what’s considered fair use is a violation of the license, for instance. In practice a human would not be able to quote exactly a 1000 words document just on the first read but “AI” can, thus infringing one of the licensing clauses.

Only on very specific circumstances, with some particular coaxing, can you get an AI to do this with certain works that are widely quoted throughout its training data. There may be some very small scale copyright violations that occur here but it’s largely a technical hurdle that will be overcome before long (i.e. wholesale regurgitation isn’t an actual goal of AI technology).

Some licensing on copyrighted material is also explicitly forbidding to use the full content by automated systems (once they were web crawlers for search engines)

Again, copyright doesn’t govern how you’re allowed to view a work. robots.txt is not a legally enforceable license. At best, the website owner may be able to restrict access via computer access abuse laws, but not copyright. And it would be completely irrelevant to the question of whether or not AI can train on non-internet data sets like books, movies, etc.