Rather than universal background checks and restrictions on specific types of firearms, how about establishing a national online "can not buy" list, that identifies those who are not responsible enough to possess them? The list would include criminals prohibited from buying weapons by law, and persons deemed mentally unfit. The justice system and mental health community would be responsible for maintaining the list.
The list would be searchable alphabetically so that the name of a perspective purchaser need not be submitted to anyone. A person selling a gun, privately or otherwise, would be required to check the buyer's ID against the list before the sale, and keep a personal record of the sale that they need to release only voluntarily, or under subpoena.
Selling a gun to a listed person would subject the seller to a severe criminal penalty, as well as civil liability if the gun they sold to an unauthorized person was used in a crime. When a gun-related crime was committed, it would be up to law enforcement to determine how the gun was obtained and to determine whether the seller had broken the law or not. If a few careless sellers went to jail or lost their life savings, gun sales would tighten up considerably.
Standards as to whom is considered mentally unfit would have to be developed, but the prohibitions against criminals owning guns are already defined.
I believe this is a common sense solution that places the emphasis where it belongs; keeping guns out of the hands of those who should not have them, rather than threatening the 2nd Amendment rights of the rest of us.