It seems to me that the US Constitution provides us the right to possess and bear arms. Not a bad idea so to protect us from the Government.
The sale of firearms appears to be under the statutes of the US and individual states and is not a constitutional issue.
Persons who desire to retail firearms should be licensed and have to do background checks on potential purchasers so to keep guns away from violent and mentally ill persons. If they do not comply, they are fined or incarcerated.
The Constitution is silent as to private gun sales, the major source for weapons for the violent and mentally ill.
Solution to decrease firearm deaths is to outlaw private gun sales and establish large fines and possibly incarceration.
QED
Unfortunately, violent criminals, and those wishing to obtain guns by whatever means necessary, have never paid attention to gun control laws, and never will.
Controlling access by eliminating private sales will help immensely.