Suppose the missile has proximity fuse. How will that change the equation?
If it is set correctly then it will detonate the warhead within range of doing damage to the target... which is the goal of launching the missile.
Obviously, if the fuse is set incorrectly the warhead will explode too far from the target is will not cause enough damage to effect the target.
The west was so fearful of the enemy finding out about the proximity fuses they were using on their anti aircraft artillery that they didn't send any to the Soviet Union during the war. Even with PF ammo their kill rate with such ammo was not amazing.
A proximity fuse does mean what would otherwise be a miss can be an effective shot.
A good example would be Igla... designed to shoot down aircraft it has an algorithm that makes the missile aim ahead of the heat source so it hits the body of the target rather than its tail pipe. this makes the warhead more effective in damaging the aircraft target.
During tests engaging Malyutka ATGMs to test the Iglas against smaller targets like cruise missiles only 5 of the 9 Iglas make kills because they didn't hit a part of the target so they missed... they only had hit and graze fuses so needed to make contact for the warhead to explode.
The result was an upgrade to the Igla-S which amongst other improvements got a proximity fuse for engaging small targets.
Unlike what is portrayed in the movies where AAMs chase planes like they are in a dogfight... AAMs and SAMs are coasting and slowing down all the time... in a turning battle with any fighter any missile would lose... the control surfaces on an aircraft are much larger and would allow much tighter and harder turns at the same speed... they also have engines opposing drag and maintaining speed... when a missile turns hard it generates enormous drag and with no engine operating to recover that energy the energy has to come from somewhere... ie speed or altitude or both.
The very small surfaces on a missile means that it was flying at the same speed as an aircraft it would stall and fall from the sky because it has no wings to generate enough lift to keep it airborne.