South Korea kinda does not care, because they know North Korea wants the entire Korea, not half-korea and half burned slab of ground.
North Korea in the past had enough non-nuclear firepower to flatten South Korea many times over, seemly they STILL have that firepower AND nukes, yet South Korea knows they won't use it...
Japan on the other hand, IS kinda paranoic, in the last elections a new party was formed, with a militaristic and nationalistic tone (including visiting WWII shrines and reacalling WWII as the good times), and they got expressive votes, also the current prime minister proposed heavily remilitarize Japan (to irritation of China, and both Koreas), including scratch the current constitution, and use a new one that follow confucionism (instead of illuminism).