This is still all pretty anecdotal and there might be truth to it - it's South Korea after all, which is a country quite known for pretty messed up values and societal tendencies - but there is still much room for questions and sources. How many are these "many parents who up the dosages" in relation to all parents? How many Korean endocrinologists who prescripe HGH to normally tall kids actually exist, and are they breaking laws/regulations while doing so? I'm aware that kids without HGH deficiency can still grow with HGH, but there's been discussions and anecdotes on this forum with posters saying the increase above the natural height potential is not more than 1-4 inches (a kid destined to be 5'5 will not grow to 6'1 with HGH so to speak).
In any case, HGH needs to be combined with aromatose inhibitors to make such a difference, it runs costly and the treatment has to be sustained throughout puberty. I also doubt that the average height of "South Korea's elite" (however that term is exactly defined) is 6'0'-6'1 already, but even if it is, I would not find the situation you described decent and worthy of imitation. Instead of succumbing to this chilling, ever-increasing primacy of capitalist competition, vanity and one-upmanship like some Asian and Western societies appear to be, should we not strive to foster a culture of understanding, mutual respect and solidarity instead, as there are far more grievous issues - such as climate change - facing us all, which cannot be solved by these forms of biological and economical arms racing?