You've given no reasonable explanation as to why there would be a war other than "NK is crazy", and "American's won't stand for a nuclear armed Korea"
When in reality, Korea isn't remotely "crazy" and no foreign policy experts who have studied the NK regime believe them to be anything but rational, if brutal actors...and the US has "stood for" a nuclear Korea for a number of years now. The genie is out of the ****ing bottle and any attack on NK becomes the catalyst for letting a nuke off the chain. It's the only way they let one loose, because again they're rational actors and their goal is the survival of the regime. If their goal was to be a nihilist destroyer of their enemies, Japan would be glowing already.
Rocket Man has to find balance between the hard liners in his military (the only domestic threat to his power), the Chinese (who would happily facilitate that transfer of power if RM goes fully off the reservation...the don't like the instability in their backyard at all) and the US. That they've done so for years suggests that infact they are completely rational actors in all of this with shitty, but entirely rational goals.