While I am sure that KIS has its merits, it does not seem to be the first choice of all of the Korean families I know.
My mother is Korean but my sister and I went to ESF schools. All of my mother's friends sent their children to ESF schools. All of the Korean children I have and am tutoring study in other international schools or continuously try to transfer to other schools. Many of these parents still care about their children learning Korean and about Korea, so send them to KIS on Saturdays for Korean lessons.
I've never sat down and had a long discussion on why they have reached this decision, thus can only speculate. One major concern they have is whether their child can learn how to speak English fluently. I think they are concerned about the English speaking environment rather than the English teaching quality. My impression is that the children, even in the English section, do not practice their English enough outside their lessons. I am guessing that this might have something to do with the student population.
Compared with more popular international schools, they do not post any examination results or record of where their students continue their higher education. Proven track record of results of education, such as sending children to prestigious schools, heavily influence parent's decisions.
However, I believe that it is still possible to send a child to KIS, teacher him/her how to speak English fluently and for him/her to end up in a prestigious university. Schools do not independently churn out model students and citizens, it also depends on the student and parents. Clearly KIS is a good option for some families.
The desire to send children to other international schools is probably further pushed by social factors, attaining something difficult or jumping on the bandwagon. The most important thing is to find a good school that suits your child, so that he/she will be motivated to learn.
Try involving your child in choosing the right school. See if you can go on a tour and observe a class together. Get your child to learn more about the school and what they offer. Provide a couple of options and go from there. Apply to a few schools, since it is quite competitive to enter some international schools.