This to me seems like the right approach, and is not much different from what humans do. Humans are free to read whatever source material they want, but you can't subsequently write it down from memory verbatim or nearly verbatim.
edit: And if you think I'm being hypothetical, Github Copilot already does exactly this.
The only possible way to do this is for companies to provide a list of all of their sources and for me to then automate verification and hope it works!
The real answer is that I should be able to control whether my information is used to train models or not, because we already know that models spit out verbatim results with generic queries and there’s just no way for a user to otherwise check this.
Regarding you wanting to opt-out of training, that's fine you can already do that for many large models. But likely doing that will become the equivalent of putting your works in a safe where no one but you will ever end up reading them or finding them.
Banning all AI training is the equivalent to banning search with any modern search engine.
edit: Also, note that you are already on the hook for not infringing patents, which if you were to try to do "perfectly" like you imply with copyright, then you need to search/read/understand the entire body of published patents. A task that is clearly not possible. Yet patent law functions (unfortunately).
What good is training an AI if nobody can use it?
Pretending that disallowing AI is the same as disallowing everyone is a strawman. People finding you is a whole lot different than your work being copied being completely okay so long as it was an AI that did it.
... or have a computer "type in" a book i.e. file copy i.e. OCR scan (or microphone for audio) (or video-record for video) ...
It's no different than a photocopier or a VCR when you're using it for that purpose, it's the end result that matters.