Of course they would remain in the system. What do you think the reason is for mandatory reporting of stolen guns!
The idea is that if you report a gun stolen, and they search your house and come up with it, you can be prosecuted for obstruction of justice.
The idea is to have all legitimate weapons under the control of the State. They do not care that much about guns that are illegal. In a couple of generations, those will be no problem, because the gun culture will have been destroyed, and guns are of little use if you do not have the will to use them, or the conviction that the right to keep and carry them is your God given right, protected by the Constitution.
Sadly, I was crossing the river in my canoe and it tipped over. Some of my finest firearms were never found.