In other words, in your example, the painter is not necessarily free to distribute their painting while ignoring the original image's copyright.
Ostensibly, they want to prevent OpenAI, and any algorithm from making works based on their work.
But, frankly, what they really want is not really that. What they really want is to prevent it from making any works at all.
https://petapixel.com/2022/12/08/photographer-loses-plagaris... is one of the more recent applicable cases here, where the original artist failed to uphold their claim, which surprised most people.
Now there are egregious examples where this is abused to target a specific artist, and I'm sure you'll find all the AI companies are entirely on board with preventing that.
But they want to produce artworks.
Note how software licences typically give you the licence to make a copy for running the program? That's because the act of copying the program from the medium (cd, USB stick, HDD...) is considered copying.
I really don't understand what's so difficult to grasp about this. And yes it is illegal to sell equipment whose purpose is circumventing copyright protections.
There isn't anything like enough space in one of these models to store all the training material, "compressed" or otherwise. Not by orders of magnitude.
It is also interesting that there are technology specific laws. You can't sell photocopiers to the public capable of reproducing bank notes for example. Or taxes added to blank medium such as cassette tapes.