Often because kids are either free labour (e.g. on the farm, working jobs to help the family, etc) or are "insurance"/"an investment" (hoping they'll make a kid who can support them with a good job in the future).

Where you don't need help tending a farm or something, child labour is illegal, and one's educated enough to know that their poor kids are more likely to grow up to be just as poor and thus not a "smart"/"safe" investment, it means that people will generally be less likely to have kids.

Most people live in cities in South Korea, where it'd usually be economically unfeasible to have kids.

I believe that education, particularly for women, tends to reduce the desire to have kids. Women without education may be less likely to envision alternative futures for themselves aside from being a wife and a mother. In South Korea in particular, it can be hard to be both a mother and a worker, particularly on the economics side of things.

This may be true, but it doesn't change the fact that birthrate is declining due to unaffordability.