I'm not sure you really need to fire nuclear missiles at other nuclear missiles as air defense. You just need something to shoot theirs down before they reach somewhere sensitive and detonate. Be it a non-nuclear missile, laser, Superman, whatever.

But the theory is somewhat sound -- you need more nuclear weapons than they can defend against. Which is a question less of quantity and more whether you can delivery payloads in faster, quieter, and therefore less interceptible ways.

Quantity is a brute-force override when you don't know how good their defenses are. Or you want enough to hit everyone in the world regardless of their defenses because you're nuts. Or you'll make someone a lot of money by ordering more, rather than better, nukes.

Honestly, the US buying more nukes is more an indicator that any previous technology gap is closing. The US is less confident in its own defensive capability relative to opponents' offensive capacity. So it feels it needs to up its offensive capacity to prevent its defenses from being tested. However, since the US is not confident of the QUALITY of its nuclear weapons program, it must make up for that with quantity.