It's more about personality than public vs. private

school issue. Korean parents tend to be more strict/critical towards their kids, by my observation. This teacher certainly has some traits of it.

However, part of going to school is to learn social skills and how to deal with people in the real world. Teachers are real people. If parents can be mature about it, it will be very helpful to the kids to pick up their learning curve. With parents' careful guidence, kids can learn through good experience or not so good experience. Best wishes!