I get where you're coming from, but I don't think even more private property is the answer here. This is ultimately a question of economics - we don't like that a) we're being put out of jobs, and b) it's being done without our consent / anything in return. These are problems that we can address without throwing even more monopolosation power into the equation, which is what IP is all about - giving artists a monopoly over their own content, which mostly benefits large media corporations, not independent artists.
I'd much rather we tackled the problem of automation taking our jobs in a more heads on manner via something like UBI or negative income taxes, rather than a one-off solution like even more copyright that only really serves to slow this inevitability down. You can regulate AI in as many ways as you want, but that's adding a ton of meaningless friction to getting stuff done (e.g. you'd have to prove your art wasn't made by AI somehow) when the much easier and more effective solution is something like UBI.
The consent question is something that needs a bit more of a radical solution - like democratising work, something that Finland has done to their grocery stores, the biggest grocery chains are democratically owned and run by the members (consumer coops). We'll probably get to something like that on a large scale... eventually - but I think it's probably a bigger hurdle than UBI. Then you'd be able to vote on what ways an organisation operates, including if or how it builds AI data sets.