Patriot PAC-3 is designed to shoot at a warhead coming down very near it. The Japanese have the Patriot. The missile was shot overhead. For that you need the Navy’s mid-course ABM system built around the Aegis Radar & Standard-3 missile. We currently don’t have a boost-phase anti-missile system as Obama canceled the airborne laser system.
More than that, however, the missile defense radars develop a target plot with a projected impact zone. If the missile isn’t projected to hit anything they let it go. That’s my understanding, andyway.
more on the airborne laser:
https://en.wikipedia.org/wiki/Boeing_YAL-1