California has the background checks in place they are pushing. this latest guy passed 3 of them, while undergoing psychiatric evaluation, while being investigated by the police.
Just as lowering the speed limit did not put a stop to all highway deaths, just as inspecting fruits and vegetables did not eradicate salmonella from out food supply, just as curbing hazardous emissions from smoke stacks did not eliminate all air pollution, back ground checks will not prevent every mad man from obtaining a gun.
But no one can deny that speed limits save lives, food inspections make our food supply safer and limiting emissions made our air measurably cleaner. Couldn't the gun community concede that back ground checks could help REDUCE the number of mass shootings at the hands of the insane? Every measure taken cannot be seen strictly through the lens of 'infringement' or 'ineffectiveness'. If indeed the gun is a benign object used by the insane to wreak havoc, shouldn't we take greater precautions to keep guns out of the hands of the irresponsible and the insane?
It's not about strictly seeing this through the lenses of infringement and ineffectiveness. It's about cost-benefit analysis.
Cost: First the obvious: Doctor-patient privelage (or at least the illusion of it that's left to us by Obamacare) is gone. Bye bye
More importantly. . .
Once the precedent is set that the "mentally ill" are disallowed from exercising their 2nd Amendment rights, it not only implies a precedent for that particular amendment, but also a precedent that one's mental state becomes a potential criteria by which constitutional rights can be summarily denied. Let that sink in. Mental state (or some bureaucrat's determination of one's mental state!) becomes a potential criteria by which constitutional rights can be summarily denied.
Mentally ill, insane, unstable. . . these are pretty subjective terms and open to a lot of interpretation. On top of that, the psychological field as a whole isn't a static thing. Knowledge of the human brain and psyche is still growing and new psychological terms are still being coined. We'd be making the concession that some person or some governing body would necessarily be in charge of deciding which conditions were enough reason to deny someone a constitutional right, and in charge of defining these conditions for legal purposes.
The entire idea of the Bill of Rights is to protect specific rights from the intrusions of our own government. The cost of what you suggest is the setting of a precedent that the government from which we protect these rights can define psychological conditions based on which to deny us those very rights.
Like it or not, what you're describing is way, -way- bigger than guns.
Benefit: Less diagnosed crazies would be able to acquire their guns legally. This wouldn't actually stop the P2P gun sales and trades, it would just necessitate taking out of the light of day those transactions made by people who couldn't or wouldn't fit the bill for a background check. Those crazies who currently get their guns through legal P2P trades and sales would still mostly get their guns, just the transactions wouldn't be legal. But some of those transactions wouldn't happen!

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So, in theory, if this were in effect since Kip Kinkle, you'd be able to stop -some- of the mass shootings perpetrated by crazies who were both diagnosed -and- acquired their firearms legally and personally. Without those guns, I'm certain that not all of them would have found some alternative method of murdering lots of people. As long as you force someone whose boiling over with homicidal urges to use melee weapons or engineer explosives, they'll probably abandon those violent desires and go join the Peace Corps.
But hey, if we can save a couple dozen people by shitting all over the Constitution, go for it. I don't need my rights. Warm fuzzy fuzzy child worship for the win! Lock me up if you have to, just keep those little cuties safe and cozy!