My dumb question: SDI and nuclear-tipped interceptors

My dumb question: It's exciting that the Pentagon's missile defense system prototype has three times in a row physically struck a dummy ICBM in tests over the Pacific. But why are we even trying to "hit a bullet with a bullet?" When you try to shoot down a flying object, you don't use a rifle, you use a shotgun or an anti-aircraft gun. So wouldn't it make more sense just to load anti-missile missiles with low fallout nukes and vaporize any incoming ICBMs that are in their general vicinity? Help me out here, folks.