> the only way to win a nuclear war is to win on surprise attack first-strike

No win: any counterstrike need only be airborne before that 1st strike hits (ignoring submarines).

In your example: NK might be able to strike SK by surprise - short distance, missiles could could cover it in a few minutes.

But in case of NK striking SK, countries allied with SK wouldn't leave NK unpunished. So even if NK's 1st strike were 'successful', it would still be annihilated by the response. If not by SK, then by its allies.

On top of that: not all 1st strike missiles would reach their enemy, hit their target, or manage to destroy all their enemies' missiles.

TLDR; there's no win in all-out nuclear war.