S - I think that South Korea (today) is likely a much more relaxed society. I have not seen any info (yet) on what complaints the Kim-Ko couple may have had with life in South Korea.
Perhaps they are like the Russians (and people of other former Soviet Republics) who are nostalgic for the 'good old days.' when you had free medical care, free education, a supplied apartment and a job for life. Perhaps you did not have much money, or much personal freedom or freedom to travel. But you were free from worries about essentials.
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And, yes you are right about the origins of much of the Japanese population, including (likely) a significant background of the Imperial family lineage. I've noted the news items of the past week, but I feel its anachronistic to call these common ancestors "korean.'
The populations of east Asia were always moving and changing. They were mobile people, moving on horseback, not usually with fixed borders. Its difficult to get a handle on the first peoples in N.E.Asia. perhaps the best mirror are the native peoples of the Americas. Perhaps twice in the last twenty thousand years there were large migrations from (what is now) Siberia, into North America, although I think there may always have been some movement across the Bering Strait, and that mobility is reflected in Japan (maybe 30+% affinity with the Mongols, and in Korea. Its also reflected in China. What we call 'Han,' is a name for many different groups of people that accepted Chinese civilisation. Often the North of China was in the hands of the Steppe people. Xiong nu and others more like the Manchus (who formed the last Imperial dynasty) and the Mongols who conquered China in the thirteenth dynasty and all the way to Adriatic sea.
Earlier in history the Huns from central Asia and thought to be part of the steppe people, contributed to the collapse of the western Roman Empire and became the nobility of both proto-France and proto-Germany.
So to speak of ethnic Chinese, or ethnic Korean or ethnic Japanese does not mean much, except in modern terms.