You are right, it’s complicated. Ballistic doesn’t mean it is a parabolic trajectory. At this distance (I stands for Intercontinental) you need to account for the atmospheric drag, winds, differences in gravity etc. Besides that ICBMs of 1960-1970s were using inertial navigation and had maneuvering warheads (e.g. SS-10), which increased complexity of the calculations. You still can do it on your modern watch.
I'm not saying the computation isn't complicated, I'm marveling at the fact that my watch can run that computation in microseconds