That wouldn't be unusal. Collection agencies are a thing and laws can be adopted to account for AI.
Then it's pay x amount of money to use generative art.
And SAAS services can be required to collect those. Self hosted Setups can come later.
You can try -- all of entertainment certainly does with DRM and all these stupids rental models for everything. They can come up with registries and agencies to handle "collections", but all of this is toothless if I can acquire the bits and do whatever I want outside of their purview.
If I can find a way to view the artwork I can do what I wish with the knowledge and emotions that I gained from it, and there is nothing you can do to stop me. This is not so different from AI learning, the exact images are not stored in the model. Neither are they "compressed". You cannot get the same exact image out that you put in.
People need to get over this awful model of rent-seeking and collections. There are always other ways to monetize.
Plus, regardless of poorly-enforced legal 'realities', you cannot control something once it is out in the public space. It may as well be public domain. Artists are learning first-hand that they have no control, as it should be.
I say this as an artist myself, either I release my work publicly and it will do with it what it wishes (paying me would be nice, but really difficult to reasonably expect), or I hoard my art privately and it dies when I die, unseen and unperceived.
The problem is the computer people have been dealing with bots for a while, we have tools and years of best practice to deal with it. Artists don't, this is new to them, and frankly a lot of them like to get wrapped up in emotions and do not give a damn about technical nuances -- as you correctly assert. So they will generally repeat whatever trite line sounds best. (You can see this firsthand on with the copypasted memes on facebook/reddit)
How many of these guys truly understand the issue? Or are they just the next generation of luddites trying to fight off the next printing press or camera?