I'm not being myopic, I'm just addressing items as they are presented to me and trying to do them one at a time. I'll try doing multiple at a time, maybe it will be more constructive.

Consider that firebombing causes far more fire than nukes and that forest fires regularly release far more soot. I'm suggesting that we do think of the scale here, and think about the logic at hand, given that nukes' primary impact is actually not starting fires or soot production. Both regular forest fires and volcanic activity produces comparable amounts of this already, and firebombing does way more of it. The number of nukes you're describing would likely produce less cooling materials than Krakatoa.

I haven't gotten to the topic of nuclear winter yet because I thought the other points might settle it first. But nuclear winter is, itself, very poorly evidenced and depends on speculation that flies in the face of what evidence we do have. It assumes that particulates in the stratosphere would hit some turning point where they stop acting like they usually do, with basically no mechanistic basis. It's just a speculative and very very simple linear model from the 80s that got turned into pop science and water cooler chat.

Nukes are still horrible, as are all the weapons we've mentioned, but nuclear winter is not anywhere near the thing to rationally fear from them.