Patriots would be next to useless in that role.  Mayhap you're thinking of THAAD, lad.  

Think about this next.  Suppose 𝒕𝒉𝒆𝒚 decided to do that to 𝒖𝒔.  Would we consider that an overt act of war?  Why yes, yes we would.  And international law would back us to the hilt.  

Likewise, if 𝒘𝒆 do that to 𝒕𝒉𝒆𝒎, it would make 𝒖𝒔 the warmongers.

But even if your suggestion had any tactical or legal merit, implementing it would only encourage NK to arc all their missiles over Japan -- because we sure wouldn't want to chance a malfunctioning or errant Patriot flattening a bunch of cherry trees in Ueno Park (and the picnickers under them).  Our Japanese allies might not take too kindly to that.