The reason why it's bad to have an age minimum in Korea for eSports is that it hurts the players, which we know from previous eSports like Brood War. In SC1, the professional teams would have extensive practice squads of 14-15 year olds that they would pay very little in the hope of developing them into top talent against the better players. If there was an age minimum in Korea, the pro teams would likely start creating practice squads of young players because they could retain talent without paying them for a starting spot. As it is now, if a player is 16 and good enough to start, like Faker and Ssumday at the beginning of their careers, they will be paid a respectable salary or they will be snapped up by another team. If an age minimum was introduced these players would sacrifice teenage years to potentially never get a starting spot if they fell off before they turned 16.
The Korean education system is also far different from the West. By the time Korean kids are in their mid-teens they have been sorted into schools of varying focuses or dropped out to pursue work. Many pro gamers come from "Digital" high schools that emphasize work on computers and so a transition at 15 to a pro team is quite easy. It's not like the American education system, which isn't focused on a specific career path so early in life.