I think this was probably more true in the past than present. US nuclear targets for the Soviet Union were declassified some time back [1]. And the Soviet targets probably looked pretty similar. And like you proposed the top priority was indeed for air superiority, because that's primarily how nuclear weapons were deployed. But ICBMs changed the game. In modern times our nuclear silos are intentionally spread out far and wide, not only in America but the entire world, and specifically designed to withstand a nuclear strike. This is without even mentioning the weapons deployed on mobile platforms such as ships or submarines.

Nuclear war in modern times is going to go for the second tier of targets: industry/economic, civilian population, and medical resources. A bomb detonated around a nuclear silo in the middle of nowhere would most likely have next to no impact on the US. A bomb detonated in the economic center of New York or California would have an unimaginable impact. The scary thing about this sort of stuff is that a "small" blast could easily have the exact opposite effect, and create a rally around the flag effect such as 9/11. And everybody knows this. So should this ever happen, the attacker is going to make sure to leave more than just a mark.

[1] - https://nsarchive2.gwu.edu/nukevault/ebb538-Cold-War-Nuclear...