It's a nice dream, but there are large practical problems with getting it to even be tried, let alone work.
The permit based system I'm talking about replaces the current system with one that ties the background check to getting the permit.
I get this. But, aside from having to go cap in hand to humbly ask the government's permission to exercise a constitutionally affirmed right, the system you propose is registering people, as gun owners, just not registering a specific ser# gun to them. And don't think there won't be such a registry. If its a govt permit, its required by law(s) to have records kept.
Then there is the fact that those people pushing the current system and its expansion won't accept anything other than their pet plans which contain the foundation for a gun registry. SO, you'd get opposition from both sides of the issue.
Next on the list of problems is the issue of state's rights. Federal law is not automatically supreme in every case. This is NOT an issue like slavery, where there is clearly a morally wrong choice. The states have a long standing, legally recognized, and vested interest in making their own decisions about some things, inside their own borders, and gun laws are one of those things, as currently recognized in our system. You, I, and the rest of the country might think those rules are wrong, stupid, pigheaded, and infringement of our rights, but if the majority of the people of that state don't think that way, they have a right to do so.
Remember that the same power and authority to say "you must" is ALSO the power and authority to say "you must NOT", and that goes for every issue you can think of.
This is the reason I oppose any kind of Fed govt top down imposition of a national carry permit type of thing. Not because being able to carry everywhere is a bad idea, but because forcing that idea on people (states) that don't want it is destructive to the checks and balances of our political system that retards and so far has prevented gross tyranny from a central government. (and note I said "gross"
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Another point against a permit system like you describe is the simple fact of time passing between getting the permit, and getting a gun. yes, most will have a gun in mind, and once they get the permit, will go get it straight away. But what about those who don't. Wait even ONE day, and a case can be made that you might have become a prohibited person after the issuance of the permit, so a current background check is still needed.
Personally, I think a better system would be to simply have the "limitless resources" of our government simply run a check on every single citizen, and put a gun/no gun icon on their driver's license or ID card.
Then, when you go to buy a gun, they can just run your driver's license number, with out any other information and find out if your legal status has changed since the issuance of the license. Doesn't need to contain any information about the gun in question either. Of course, in order to work correctly the system would have to be created within the government to ensure timley and proper updating of your "citizen license" and of course, there is a potential for abuse...
still boils down to a degree of registration though. If our govt is involved in any way, there will be some kind of record, somewhere. If it ever gets to the point of them rounding up gun owners, the list of people who applied for a permit is going to be their starting point, and it doesn't matter if you don't have any guns when they come for you, or didn't even get a gun at all, your name will be on their list and you'll be in the camps with the rest of us, if you survive the arrest....
don't stop dreaming, though. However unlikely sometimes dreamers do change the world.