Knowing who owns what gun does nothing to hamper any illegal gun trade. Criminals don't register their gun purchases. Nor do they submit to a background check before they steal a gun or buy one out of the trunk of a car.
Precisely. Gun registration tracks guns of the law abiding, not criminals. It provides no benefits to identify criminals after a crime and states/nations (links Canada) that implemented registries often end up dropping them due to low benefit and high costs. They have their greatest "benefit" at confiscation time. It is a thoroughly evil idea and anyone that supports it is no friend of the 2A.