Sorry, But the Numbers Do Not Work

The USAF is looking at an air launched missile to intercept ballistic missiles during boost phase. (paid subscription required, but you can find a shorter version here)

Raytheon is pitching something called the Network-Centric Airborne Defense Element (NCADE), a, "two-stage missile with an extended-range Amraam solid-fuel rocket motor as the first stage, a separation joint, and a second stage with a divert attitude control system and guidance-unit electronics and an AIM-9X seeker."

It's unclear to me what the range would be, but an AIM-9X (Sidewinder) missile seeker head does give us an idea as to the maximum speed of the missile: about Mach 2.5, about 850 m/s, because of the hemispherical seeker head at the front of the missile. (see picture of AIM-9X forebody courtesy of Wiki)

Beyond that speed, the drag, to say nothing of the heat generated, would be excessive.

It's why higher speed radar guides missiles have a conical or ogival nose cone.

Well, if we work with the basic information for the launch performance of a Minuteman ICBM, 2 Gs at launch, this gives you 42 seconds until the ICBM is flying faster than the interceptor.

In reality, the time for the ballistic missile to reach mach 2.5 would be less, as acceleration increases as fuel is burnt.

This is not a viable interception time unless you know exactly when and where the launch is occurring, and you are close enough to make the intercept within 30 seconds from launch, about 25 km.

Also note that the Minuteman is an old missile, and newer designs, such as the now retired LGM-118A Peacekeeper, flew out faster, and were ejected from the silos with an expulsion charge at a not insignificant speed, so the interception time is realistically far less than 30 seconds.

This really does not sound to me like a viable platform.