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" You have found the answer, it is incredibly expensive to raise children in the ROK. It is not unusual for parents to budget 40 or 50% of their take home pay for their children's education, the largest part of this going to hogwans (private institutions) or tutors that prepare their children for the multitude of placement tests that determine a students educational future.
Families are also having children later in life than they were before, so naturally they have less time for children.
There is also a strong preference for sons, (the reasons or many and cultural and I don't want to get side tracked) so many families will only have one child if their first child is a son. Many parents who have a daughter will have another child late in life in an effort to have a son.
The divorce rate is rising so many mothers who would have tried for the second child later in life do not because they are now single.
The exodus from the country side in driven by women and children. On average women much prefer city life to farm life, in fact many farmers are marrying Philippino or Chinese women and bringing them to Korea, because they cannot find Korean women who want to live in rural areas. If you want your child to have a chance to get into the best universities, Seoul National, Yonsei, Ewha etc they need access to the best institutes and private tutors these, of course, are found in the big cities.
These are the largest reasons for Koreas declining birth rate and the move from rural to urban areas. "

Same thing is Japan ...