The most interesting phenomenon of nuclear-tipped ABM systems is that exoatmospheric nuclear explosions create a pulse of X-rays that travels through space until it hits the atmosphere where it creates a plasma explosion that creates a strong and dangerous electromagnetic pulse.

Aside from the effects on people 1000s of miles away (the plasma travels that far) the EMP is a direct threat to the radar, computers, communication links, etc. In an engagement that lasts a few minute it needs to recover from any upset quickly; note Spartan uses a track-via-missile system (common in anti-aircraft and anti-missile systems that engage fast and distant targets) which would also impair missile control during an upset.

To that end they did all kinds of work including developing a unique computer

https://srmsc.org/ref1040.html