Astute of you to factor in the South Koreans into the debate. The problem of North Korea is not Americacentrtic. What do the SORKs want and how do they go about achieving that goal?
The ROK wants the north’s military threat gone, but at the same time don’t want to have to face war with their brothers. They dream of a peacefully reunified democratic Korea (although the younger generations increasingly care less about reunification than the older generations). Liberal ROK leaders generally try reaching out with olive branches, which the DPRK exploits. For example, they’ll send aid, which the DPRK never tells their people that it came from the ROK, and instead pretends that __they’re__ the ones sending the __ROK__ aid. At one point the ROK built a joint industrial complex in the DPRK; just a couple years later relations worsened and the DPRK seized the complex; now they’re looting all of the buildings and selling the hardware from them. It’s the same story every time.
The ROK tries to maintain enough military threat (including alliance with the US) that even if the DPRK launches a very painful initial attack, it’s ultimately Pyongyang that will fall, not Seoul - thus preventing one of the main national goals of the DPRK, reunification of Korea under the control of the Kims. The Kims will not give up on their long term goal, however, and they know that ultimately it requires getting the US to abandon them. And that this requires ICBMs and thermonuclear weapons that the US can’t stop, in order to blackmail the US into leaving the Korean peninsula - knowing that the US will always be more afraid oflosing its cities than they are afraid of losing their cities. So they’re not going to abandon their nuclear and missile programs, no matter what the cost.