There's nothing in there about the Air Force either. You missed the commerce clause gives Congress the authority to regulate the sale of firearms. That means they can license dealers and creat regs regarding participants in any particular area, or item in commerce.
The fed 18USC922 gives a list of things that should disqualify folks from partaking in firearms transactions. As it is now, the States are requested to submit info regarding fed disqualifiers to the DOJ for entry into the NICS database. It's voluntary per 28CFR25.4 now.
In 2002 the house passed a bill to change that to must supply the info. Had that bill been passed by the Senate and signed by Bush, Cho's court record would have shown up and hte firearms transaction would have been denied. That would have been a good thing.
The Bill of Rights limits the scope of the Constitution and specifically excludes any infringement of the right to keep and bear arms from the government's power to regulate commerce.
If it is so "reasonable" for the federal government to regulate arms, why did our Founders prohibit it?
If you don't have enough laws already, when do you think you will have enough? The slippery-slope suggests that it will eventually be suggested that people have to undergo psychiatric examination in order to purchase a gun. Why wait until commitment is mandated if a timely examination can reveal the problem?
Do you think Cho was a suitable candidate to own a gun on the day BEFORE he was institutionalized? I don't. But with freedom there is risk. There always will be.
And before you suggest that my slippery-slope argument is invalid, consider that Kalifornia is not "shall-issue" with respect to concealed carry permits. One of the requirements in my county is an interview with a "counselor".