The key word there is if. If you catch a criminal with a gun, he'd usually be holding it illegally within pretty much any current US law. The problem is that catching those criminals before they do harm is problematic - because until they do, you don't know who they are. And doing random searches on law abiding citizens just in case they are hiding something would probably be too much even for Brady campaign.
Evidence shows that in countries where private possession of guns were banned - like USSR - the criminals did have guns and did use them to kill people. Government restrictions rarely make something wanted inaccessible - prohibition failed miserably, war on drugs did the same. War on guns would inevitably fail too, but would bring enormous amount of victims among regular citizens - just like the ones mentioned above did.
Evidence shows that in countries where private possession of guns were banned - like USSR - the criminals did have guns and did use them to kill people. Government restrictions rarely make something wanted inaccessible - prohibition failed miserably, war on drugs did the same. War on guns would inevitably fail too, but would bring enormous amount of victims among regular citizens - just like the ones mentioned above did.