It depends on how the US and South Korean forces go about rapidly destroying all the North Korean artillery pieces already in place. Yes, there will be a level of South Korean casualties, I agree with your analysis to some extent.

This would have to be done extremely swiftly and by surprise and obviously using multiple waves of fast moving fighter jets. Then destroy all their airfields and jets with another wave.

That would be the first thing to do. And very shortly after hit the leadership with those MOABS and assorted bunker busters.

The thing is, does the North Korean military really have a serious nuclear capability? If the answer from the "experts" is yes, then they will need to be
taken out regardless of any collateral damage. We can't risk a nuclear exchange from a man who is totally nuts.