It was driven by the desire to be able to guarantee that, even if the Soviets attacked first and took out as many military and infrastructural targets as possible, the US would still be able to launch a crippling nuclear counterattack. The Soviets wanted the exact same thing, but in reverse. It wasn't driven by bragging rights; it was driven by the dangerous fact that the advantage in a nuclear war goes with whoever strikes first.

The development of ICBMs, missile submarines, and so on, actually made us a lot safer by making the "strike first and strike hard" approach much less effective. The situation was more volatile back when it took a month or so to put together the bombs for a nuclear attack. When you look at it from such a ruthless perspective, the Cold War made a weird kind of sense.

And as pointed out in Wargames, whoever strikes first still loses.

MAD (mutually assured destruction) is not a pretty thing.

The point is to make it so that whoever strikes first still loses. If the first-striker advantage is big enough, then the country that strikes first could survive with their population, government, and military capabilities largely intact, while pretty much destroying the other side.

I think that the scientists behind the US research effort just had a better budget.

Bragging rights are directly correlated to budget, and maybe even vice versa ;)