To properly intercept ICBMs, you have to intercept them in the boost phase. Once they're in the re-entry face, interception becomes much harder, as you have 12-16 times the number of targets, and they're actually more maneuverable than the actual missile. Half those targets aren't even equipped with warheads, so intercepting them is pointless - except you don't know which is which.
Closing speed is hypersonic. Warheads are hardened against EM pulses and atmospheric re-entry. The most likely means of interception is a close-proximity nuclear explosion (within a hundred meters, for best results). This is likely to trigger a massive EMP, and will take off-line your missile defense system. With conventional warheads, the required precision is much higher, I would wildly guess within a few meters, at most. This requires precise tracking, guidance and high energy reserves for the closing move, to counteract evasive maneuvers. Also, your system has to be resistant to enemy EMP, specifically the warhead, but also the ground stations for guidance, tracking and control.
The problem with boost-phase interception is, that the interceptor has to be in the air before the launch, or be launched simultaneously, from very close by. This almost renders it equal to an anti-force first strike, as your assets have to be in position and on a hair trigger, to intercept. This requires massive reconnaissance effort, to cover all launch sites, and a massive air- and sea-borne force, in permanent readiness and extreme proximity to the territory from which you expect launches.
In the case of NK, this is rather feasible, if they had a significant ballistic missile capability, but in the case of China, it would be impossible to pull off.
This is, why it's so tempting to put your warheads into orbit during peacetime. You can use them both to intercept ICBMs during the boost phase, and you can drop nukes on your enemy, without having to actually launch anything.
Also, comparing an ICBM intercept to a Scud intercept is rather generous. Scuds are "slow" "low-altitude" medium range (<2000 km) missiles, that give plenty of warning time, and face atmospheric drag for most of the flight phase. They're equipped with single warheads, and I suspect have low terminal maneuverability. Their warhead is designed to detonate on/close to impact, where a MIRV will usually aim to detonate at 200-1000 meter altitude. a Lucky CIWS might be able to take a Scud apart in terminal, but I'm not sure if the system can even track a Mach 8+ MIRV.
Similarly, most tactical intercept systems probably scan for launches inside the theater, and are unlikely to even detect a MIRV, which will usually approach from around 45°, at "radar infinity".
The flight envelope of a MIRV basically prescribes the use of a dedicated defense system, and the demands to those are barely being met by current implementations.
On the other hand, defending against significantly sub-orbital missiles with only single warheads, is significantly easier, especially if they require a direct hit to inflict the required damage.