I've read your comment 3 times, and I'm still not exactly sure what you are suggesting, or what the purported benefit is? (it's probably my poor interpretation, but I'd appreciate if you wouldn't mind expanding :)
The current solution is to download the same thing twice. It is wasteful and I offer downloading it only once, but if site A asks for a resource for site B, the resource should not be returned immediately, but after a few ms so that site A gets the wrong conclusion.
It should work if site A can observe only the result of the request but can't check if a request happened or not.
He would lose performance but win more privacy.