> everybody else can be relied upon to complete the job once started--so the logic goes

How does that logic goes? Is it some simplistic tale we tell children or an actual strategic calculation?

> so you can start the chain reaction that ends all human civilization

You know that the goal is not to end all human civilization?

> it seems like the best move would be to have far fewer nukes than everybody thinks you do.

It sounds like you haven't heard of the nuclear sponge and the shell game. Let's say you want to hit your adversary. And because you are not suicidal you want to do it in such a way that they can't hit you back in retaliation. "Simple!" You just aim a warhead at each of their rockets. Now how many warheads do you need to do that? About the same number as many rockets they have. One for each. Or maybe 2 if you are accounting for malfunctions. Or maybe 4 if you want to saturate their defenses.

And because you can put multiple warheads on a single rocket this favours the attacker. Because the defender needs even more rockets to "soak up" the warheads of the attacker. And the defender needs even more than that so they can ensure a retaliation, which in turn stops the attacker from even trying in the first place.

So that is why you need many missiles. This concept is the nuclear sponge.

And of course this is super weapons race-y. Because if you need more missiles than your enemy, and your enemy need more missiles than you... there is a problem with that.

So what you can do is that you build more missile launchers than you have missiles for. And you play a shell game. You have X launchers but only 40% of them are ever armed with a missile at any single time. And to prevent your enemy figuring out which ones are filled you play complicated distractions. For example you make sure the empty launchers are guarded just as much as the full ones. And you periodically go with a big truck to swap missiles around. Or maybe to swap nothing with nothing? Or to bring missiles to maintenance? Nobody knows, and nobody is supposed to know. If you do it well enough your enemy won't be confident in their knowledge on which launchers are filled so they have to hit all of them if they want to be the one who first throws a punch.