We don't let felons own guns and the mentally unstable probably shouldn't either. So the problem seems to be the lack of a system to determine that. Just how exactly are your right to arms violated with a mental/criminal background check?
Rabbi addressed that in much the same way as I would, so I guess I don't get the point of your question.
Some states do allow convicted felons to own guns, by the way, as long as they've fulfilled all the terms of their sentences, though, to my knowledge, none of these states will issue them permits that would allow them to carry concealed on government property. Also, other states provide for a process of review by which ex-offenders convicted of non-violent crimes may have this civil liberty restored.
But let us get down to the ultimate reason why many of us oppose the notion of prohibiting persons to own guns on the basis of past mental illness that is not the stuff of real psychosis: we don't trust the government to abuse that standard on the basis of a political agenda.
We see the writing on the wall.
You do understand that the ultimate essence of the right to keep and bear arms is the right of the people to resist usurpative government, right?
I don't have a problem with prohibiting schizophrenics from owning guns, but that condition is so rare as to be virtually moot, compounded by the fact that most schizophrenics are obvious. They're not going to walk into a gun store and walk out with a gun. And I don't know of any previously diagnosed schizophrenics who were responsible for mass shootings, do you? Violent schizophrenics are a very rare breed. Most schizophrenics are not violent at all, just hopelessly dangerous to themselves without medication or supervision.
In most of the instances in which schizophrenics attacked others with guns, they were diagnosed as such after the fact, as the dangerously violent ones are
paranoid schizophrenics, most of whom, ironically, are reasonably functional from all outward appearances and socially isolated; and, by the way, even most paranoid schizophrenics are not dangerously violent either.
Generally, mass shootings are not perpetrated by schizophrenics; rather, they're perpetrated by ideologically driven or grievance-driven narcissists/sociopaths without previous criminal records. They're morally insane, not clinically insane. Only in very rare instances are we going to know who these people are, let alone have any actionable means of prohibiting them from owning guns, until after the fact of their mayhem. In the meantime, the best defense against them is a law-abiding, armed citizenry.
So based on this fallacious line of reasoning, we're just not willing to give an inch to those whose real agenda is to disarm us all.
Hope that helps.