I think it would. A check-in check-out system would help.
But I will concede that Stoked is right in that it would still take a long time to come into full compliance. Even if, somehow, the government got away with the obtrusive door to door searches.
I doubt this would ever have full compliance. There are millions that will openly defy this from day 1. This is one of those issues I can see causing a split in the nation. As a group the south, great plains and most the rocky mountain states (CO and NM are possible exceptions) would refuse.