See, this is a problem with the usual description of the present perfect as meaning some action that started or was completed in the past but has relevance in the present. This kind of description tries to get the idea of the present perfect but fails to. And your question is a good example of it.
Just about anything that happened in the past still has relevance in the present.
But grammatically the present perfect cannot refer to an action or series of actions that was entirely completed in the past. Break out means begin and in this context, a war can only begin once. If it stops and then starts again, we would say something like begin or break out again.
Since the Korean War had only one beginning, and that beginning took place in the past, we need to use the simple past tense.
The Korean War broke out in 1950.
Obviously, it is true that this past event is still relevant to the present. Neither side has signed a peace treaty and the war has not technically ended. But it broke out (started) once, and that was entirely in the past.
But you cannot refer to an event or series of events that took place entirely in the past with the present perfect, because the present perfect includes within it a reference to the present moment, the actual moment of speaking, not to present time in general.
You can say:
The Korean War broke out in 1950 and has continued to this day.
The present perfect can be used because the action of continue began at a point in the past and is still happening now. But the action break out is over.