https://en.wikipedia.org/wiki/Nuclear_winter suggests more widespread effects as at least a plausible scenario, though not by any means a certain one.

That aside, widespread radiation would ruin significantly larger areas (nuclear test sites intentionally avoid areas and conditions that would allow the spread of fallout, whereas bombardment would not); destruction of numerous functions of society and common services would kill many more.

Whether some subset of humanity would survive or not, this at the very least seems like a scenario well worth preventing.