Such a scheme does not work without cookies in the first place, so the optimal strategy for scrapers is to keep any (likely multiple) session cookies until they expire. Technical details aside, if a site becomes a worthy target, a scraping operation running on billions of dollars will easily bypass any restrictions thrown at it, be that cookies, PoW, JS, wasm, etc. Being able to access multiple sites by bypassing a single method is just a bonus.
Ultimately, I don't believe this is an issue that can be solved by technical means; any such attempt will solely result in continuous UX degradation for humans in the long term. (Well, it is already happening.) But of course, expecting any sort of regulation on the manna of the 2020s is just as naive... if anything, this just fits the ideology that the WWW is obsolete, and that replacing it with synthetic garbage should be humanity's highest priority.