There is a difference between advocating something and merely believing it. But I'm mostly skeptical of the people that put "strongly disagree" on that question. As opposed to "disagree" or "neutral". The fact that it's so correlated with political ideology is more evidence that it's just political bias.

This correlation is what interests me - it does fit with political bias, but could it also be that the political views are a product of beliefs in HBD?

I really don't know. When I researched this it seems like the effects are pretty hard to estimate. Different models give very different results. A recentish study using more modern climate models shows that the effects would be catastrophic and last for multiple years:

I think I might have put too much faith in one study. Perhaps 90% deaths is plausible.

Those first few months are the problem though. The crops and livestock die or absorb the radioactive isotopes. The people too if they don't happen to have a fallout shelter handy.

The crops may absorb the isotopes, but the isotopes will continue to decay, and so by the time the crops are to be eaten they should be fairly safe. I agree that there would be terrible casualties, but I don't think it would be as bad as having to spend a decade underground.

Moreover, my main point here is that the cloud of radioactive death might kill 95% of the US/Russia (or whoever the primary belligerents are) but by the time it reaches Brazil for instance it would be a lot less radioactive.

If that happened in the modern world, technological civilization might end and never be restarted.

I agree. The end of technological civilization is a different point from simple mass casualties - if 'only' 40% of humanity dies, but those 40% are concentrated in first world countries and urban centres, would the survivors be able to rebuild? Machinery would continue to work for a while, although the oil distribution chain would break for a while at least, but in the long run machinery would break. The factories tend to be in the first world countries that have been nuked, the universities in the cities have been mostly destroyed. Moreover there would likely be a general luddite tendency to blame technology for the crisis. Its probably easier to restablish resource extraction then to restart scientific research, and so we would be less likely to develop renewable energy before the fossil fuels run out. I suppose the end of technological civiliseation would reduce the population back to medieval levels, although this would be a long process of resources slowly running out and machinery slowly degrading.