Kim's stated reason for nukes is to preserve his regime. Which is actually reasonable. I don't recall the exact quote, but it has been said the US does not treat nuclear armed nations it disagrees with the same as non nuclear armed ones. Kim states worry over South Korea, with US backing, overthrowing his regime. And I agree that he realizes he can be nuked in return. What I don't know is what (outside of direct South Korean invasion) Kim would interpret as a threat to his regime. Nor do I know his reaction to a nuclear exchange between the US and China happening on his doorstep. Would he consider that a prelude to US invasion and decide to prempt? How allied to China is he? (I suspect not actually in any really friendly way. Geography has forced him to cooperate.)

I completely disagree with your certainty that "There is no way China could ever successfully complete an invasion of Formosa across the Taiwan Straight." We have wargammed a healthy number of scenarios, and China has won more than a few. I am sure that in their own wargames, China has won quite a few more. Nothing about war is predictable. If China feels confident enough at all at any point, your "certainty" means nothing. There were voices within Japan that warned of disaster in a long term war with the US. They were drowned out at the political level, and Yammamoto hoped he would strike enough of an opening blow that it would not be a long term war. If China decides to attack (however irrational that may seem to you) and the US decides to respond, then two nuclear armed powers are in open war with each other. The possibilities for misinterpretations and accidental nuclear engagement alone are several, and escalation of any sort is a monster which drives itself. Leaders often feel they have "no choice" but to keep escalating once it starts. Some movies show sanity prevailing at the last second. Others are darker. Both are valid outcomes.

My study of history is not optimistic about the nature of my species. Our capacity for war is intense, and once two big powers wind up and go at it "all bets are off." I don't see many "happy endings." There are indeed a few. They are outnumbered.