I don’t think S. Korea is really looking for the United States to start a war in their region. Their capital is going to get destroyed in any such conflict.
I served in Uijongbu. We regularly joked about how we served as a speed bump for N. Korea so that if they dared to do something It would only served to draw the American government into war if N. Korea because they dared to mess with our miniscule force presence.
The point of the exercise is to bully the Iranians into a response that can be used as an excuse for the war that factions in this country desire in an almost sexual way.