The things you listed are already in place, and have been in place for decades, though the age requirement differs by state. For example, in Florida, it's 18 for a long gun and 21 for a pistol. There is often a 3 day holding period, unless you own a license (concealed carry or firearm permit). You cannot buy a gun from any retailer or FTL without a background check. You also have to fill out a firearm transfer form to purchase the gun, which also registers the serial number to your name.
The only part where it breaks down is private sales, but the person selling the firearm can be held liable if they end up selling to someone who cannot legally own a gun. The same scenario of registering the sale applies to private car sales. You can sell a car to someone, and if you don't inform the DOT you've sold the car (and the buyer never registers it), you're on the hook if they find the car abandoned on the side of the road or at a crime scene.