If a single ICBM can destroy a city, why were they launching hundreds of nukes? Isn't that No Kill Like Overkill?
Cities aren't the only targets, you also target the other side's missile sites. These missile sites are dispersed so as to require individual targeting. This means you need at least one nuke for every nuke the other side has, plus all the others you need to hit other targets. The other side, of course, has the same philosophy, so they deploy more nukes to hit your nukes, you deploy more nukes to hit their nukes, and so on and so forth. Throw in ballistic missile submarines which can take out your nukes but cannot be themselves targeted in return, and your grand strategy eventually becomes "rocks fall, everyone dies".
If you're going to launch the nukes, drop that big a hammer, then make sure they can never get back up. A similar rationale is used by cops for "double-tapping" when they resort to lethal force.
Also, the sheer number of nukes alone was intended as intimidation, as part of Mutually Assured Destruction — essentially sending the message to the other side that no matter if even half their nuclear arsenal was destroyed in a first strike, there would still be enough to ensure the complete annihilation of the enemy.
Not just overkill. Planners had to assume that a large number of missiles would miss their targets (Against "hard" targets like ICBM silos, strategic command posts, and some megastructures, even a nuke has to hit very close), be shot down by SAMs (exceptionally difficult, but possible), or simply malfunction (rain, manufacturing flaws, blast from other warheads). After the Soviet Union fell, documents were obtained suggesting that Soviet planners assumed a thirty percent failure rate, and a high number of misses. That's why they targeted so many missiles in the first place.
In addition, Soviet strategists were convinced that the US intended to launch a first strike against Soviet missile bases at some pointnote In a sense we did have such a policy, in that we never renounced a first-strike posture in Godzilla Threshold situations such as the one presented in the film— the Soviets had, at least publicly, so they built massive redundancy into their targeting systems. Kansas City itself was likely targeted in real life by fifteen or more missiles, all originating from different launch sites, and each target - Whiteman Air Force Base, the ICBM sites, the Kansas City Plant - was likely similarly overtargeted.
In one scene Jim Dahlberg's horse is dead, but in the next he's driving a wagon and four. Where did those horses come from?
This and other apparent errors were caused mainly by Executive Meddling. The network had originally wanted the film to be four hours long so it could run on two successive nights. The director didn't think that breaking up the show at the point of the attack was a good idea and wanted it to run 2 1/2 hours on one night, but he dutifully shot the whole four hours. When it became apparent that the network couldn't sell any ads for the post-attack section, however, they requested the director to cut the film down to 1 1/2 hours to minimize their losses. The director finally negotiated a two-hour runtime, but in doing so he was forced to cut a number of scenes, such as:
a neat special effects scene showing the destruction of Kansas City from the cockpit of a 737;
a scene showing how the hospital got its radio;
the immediate effects of the blast on the University of Kansas, including how Joe Huxley's glasses were broken;
most of the backstory of the Hendrys, Sam Ichiya, and Airman McCoy's family;
how Jim Dahlberg got the horses, and the identity of the old man they found lying dead in their kitchen;
the reason for the ribbon Denise Dahlberg was wearing in the gymnasium (it was a triage sign meaning no chance of survival)
Nurse Bauer's death scene
Really, why are they using hair ribbons as a mark? How did that conversation go? "Hmm, how should we identify people with no chance of survival, many of whom are losing hair? I know! How about a hair ribbon? That's makes total sense!"