You can't hit what you can't see.

Put an ablator on the nose of the ICBM. As it approaches burn-through the warhead salvage-fuses. Your defenders have EMP issues, but let's assume they can shield against that. The problem is now the sky is full of electronic ghosts from the first detonation. Pretty soon another warhead comes along through that ghost-filled sky. The defenders have a much harder time locating it accurately enough to fire. And, once again, when it's going to burn through it salvage-fuses. The sky gets worse.

You can stop any given missile. Stopping a whole string of them is quite another matter.

(Same as an aircraft pilot beating an incoming missile. Do it right and you can use the missile's speed against it, forcing it to make a turn it can't. But now you're out of position and can't do it against a second missile coming in some seconds behind.)

This was one of the ideas pursued in the Reagan-era Strategic Defense Initiative, aka Star Wars. They weren't able to develop a system that would work against ICBMs. Powering an aircraft is both easier and harder; the aircraft presumably isn't making evasive maneuvers and trying to stop you from powering it, but the economic and safety constraints are harder.

IIRC, one of the problems with SDI was that at the time the only way to make lasers powerful enough to destroy an ICBM was to pump the lasing medium with a nuke, making them one-shot devices.

Sensing may still be a problem today, especially as stealth is also improving, but detection in general is much easier than it was in the Regan era.