I am no expert on the reasons for these airports but could it be something to do with South Korea's rather friendly and welcoming neighbour from the north?

Forget the swanky terminals but the airfields and runways could come in handy should war break out and the Americans/South Koreans need airbases.

Better to have loads of airfields all over the place rather then a few. Makes it harder, more difficult and more annoying for the North Koreans to destroy.