The flaw to the private transaction background is its not enforceable. Unless your talking firearms made at the time of the making of the law.
Retroactive would be impossible. Ive bought and sold guns individual to individual for decades. Traded, given, been given firearms. I couldnt possibly remember who they all were i traded sold etc with.
All one has to to do circumvent the private background even for a slightly older gun in question would be to say I have that gun for years I dont remember who I got it from. End of story.