It's not like we don't know what does or does not work. We can just look over the border at Canada, which also has a high number of guns per populace, but much much lower incidence of gun deaths.
The control of firearms in Canada is predominantly governed by the Firearms Act, the Criminal Code, and their subordinate regulations. The Criminal Code defines the main categories of firearms, which include restricted, prohibited, and non-restricted firearms. The Firearms Act regulates the possession, transport, and storage of firearms.
Applicants are required to pass safety tests before they can be eligible for a firearms license. Applicants are also subject to background checks, which take into account criminal, mental health, addiction, and domestic violence records.
The Criminal Code identifies “the various firearms, weapons and devices regulated by the Firearms Act.”[4] The Code classifies firearms into three categories: restricted,[5] prohibited,[6] and non-restricted.[7] Non-restricted firearms “include ordinary shotguns and rifles, such as those commonly used for hunting. But some military type rifles and shotguns are prohibited.”[8] Restricted firearms include “certain handguns and some semi-automatic long guns (not all semi-automatic long guns are restricted or prohibited). Rifles that can be fired when telescoped or folded to shorter than 660 millimeters, or 26 inches, are also restricted.”[9] Prohibited firearms “include most 32 and 25 caliber handguns and handguns with a barrel length of 105 mm or shorter. Fully automatic firearms, converted automatics, firearms with a sawed-off barrel, and some military rifles like the AK 47 are also prohibited.”[10]
Note also that “antique firearms are not considered firearms for licensing and registration purposes.”[11]