In America, full-auto firearms and sawed-off shotguns are registered under the National Firearms Act, and the registry was closed to full-auto in 1986 by the Hughes amendment to FOPA, which means remaining machine guns on the civilian market are getting rather expensive. If you use registration as a tool for attrition, you create a two-tier system where the rich own the most powerful weapons and the poor don't get none. That's not how our Constitution is supposed to work.
I'm all for keeping the NFA in place (but killing the Hughes amendment), mainly because of leveling the playing field on labor issues - in the Battle of Blair Mountain, corporate mercenaries used full-auto weapons on striking coal miners, including at least one machine gun mounted on a tripod, judging from the number of .30-06 shell casings found at the site and their circular dispersal pattern. Most miners were lucky to be carrying a shotgun or a lever-action rifle; most of them couldn't afford a Thompson. This happened a few years before the NFA was passed, though.
When you buy a firearm, you fill out a Form 4473, which also records the serial number of the gun you're purchasing, so there is a registration system in place right there. We could stand to tighten the background check system, sure, so that means much fewer private sellers dodging Form 4473 and a better accounting of which guns are being sold to whom.
But if registration is meant to lead to a permitting system like the Illinois FOID, I can't support that as being Constitutional. The system has experienced at least one catastrophic failure, which means someone couldn't buy a shotgun in Illinois because Windows XP crashed. If you're going to deny a firearm purchase, do it for the right reasons.