hah, mostly very good points
IDK, for one I'm not even sure a major exchange is as deadly as here supposed (much depends on who else joins the party I guess); sure lots of land on some continents at least is badly irradiated, but a majority is not quite as bad, and most of radiation degrades quickly, some simple measures help, and besides not everyone needs to survive every year After Launch anyhow, and non-extreme radiation doses kill only statistically. Much depends on - as you say, on how much of state command structure is able to survive and organize the remains, and that could easily vary from state to state on the planet.
I lived through a minor war in my youth, with the frontline maybe 2km at its closest approach and regular shelling for IDK couple of years. It was a remarkably well-ordered affair, considering. Fact half the GDP evaporated and rest was put under direct government command for war and other logistical purposes didn't really constitute a panic collapse of societal order or anything like that, and return to some kind of (low budget) normality quite quick. Kinda remarkable in retrospect, in how badly we react to small upsets, like a high-single digit GDP loss in a recession, yet tolerate such major disasters...
If the number of people directly killed is on the order of a few hundered million, thats not a substantial part of the humankind, so it may not be too different -- to people living in places too boring to have been hit by a nuke and not too close downwind from something interesting, ofc.
Guess it becomes worse if all the players hit all the other players, and no further advances in reducing nuclear inventories are made before it hits etc.
But again, yeah, a societal collapse is certainly a kill-myself kind of event for me too, because as you say - we're never gonna rebuild if we fall to that point. I'm just more sanguine about nukes being lobbed about not necessarily causing this I guess.