And I am sure adversary will use thermonuclear bomb to attack cities where everyone has been in the bunker, and refrain from using air bursted dirty bomb.Keep in mind that if there is an incoming first strike, the adversary has most likely evacuated their population and military assets. They also have a smaller nuclear arsenal now. You may as well take advantage of it and wage nuclear war over a long time span.
The problem of "winnable nuclear war" is the false assumption of your enemies being reasonably and playing by "rules."
First strike against hard targets won't bring anyone to the negotiation table but diplomatic catastrophe. "OK we just disarm your silos and we know you guys still have TELs on the road. Could you please come to negotiate with an unconditional surrender?" The idea of surrendering is plain stupid when you still have capability to second strike adversary's soft targets. And the man, who just put entire mankind at risk and thought it is a W not a big fat L, probably should call a psychiatrist first.
And a first strike against soft targets (w/ or w/o hard targets) is even more stupid. Because hard targets are deep inland and will get an early notice to launch on warning with thousands of warheads.
Nuclear war is only winnable when you can disarm at least 80% of fixed delivery systems AND your adversary is rational. Neither of them is true under a nuclear war scenario, especially in 21st century. Disarmament is impossible with a credible LoW posture. Russia can't even disarm Ukrainian Su-27/Mig-29s in several days, how is it possible to disarm nuclear forces in several minutes?