[0] https://cyber.harvard.edu/property00/MP3/rio.html
[1] https://www.tvtechnology.com/news/appeals-court-sides-with-d...
[2] https://www.eff.org/deeplinks/2020/11/github-reinstates-yout...
Unfortunately, the way they've historically done it is to try and block anyone with force until someone came along who actually figured out a peaceful model for both consumers and producers (again, think Spotify). I don't think the AI game will play out any different.
The cool thing is: it's a great place for startups. Big tech will get sued right away before finding product market fit with an AI product, but small startups can survive until they have enough product market fit to fund the defense for the lawsuits.
But I disagree, I think they can stop it, at least for a long while, because of the enormous expense that it takes to train and run these models (at least for now). Nobody operating in legal gray or black territory could afford to do this. If they use the force of law to basically block legitimate uses, it will slow progress to a snail's pace.
Meanwhile they'll use the tech to stay way ahead in the cat and mouse game, so when somebody does generate something they can block it/censor it right away before it's able to spread. That will be good enough for their purposes.