Sure. Some idols you mention are actual citizens of a country other than Korea. They just work in Korea with a work visa. Work Visa's are temporary permits that allow the individual to make money, live, and anything else they want to do in Korea more or less. They are almost but not legally citizens of Korea. Those people since they are not citizens of Korea do not have to register with the military and are ineligible from serving.
Other case may be citizens of Korea but have never resided in Korea but lived their lives outside of the country. They were born in Korea so they are technically Korean citizens but shortly after that they moved to say Great Brittan. They don't really travel outside of GB so they have no passport or anything and are just residence holders in GB. Those also are not to be drafted even though they are citizens. They are not forced to go to Korea and join the military.
Similarly if you are a Korean citizen but have a passport from another country (say refugee status) and not one from Korea then you are exempt from military service in SK (until) you reside for 3 years inside Korea. If you stay longer than 3 years you are then required to join the military.
Say you have a dual citizenship of GB and Korea for whatever reason and have two passports for whatever reason. In that case because you have a passport from Korea you will be drafted UNLESS you renounce your Korean Citizenship and retain only your GB one. This actually happened and the person was blacklisted from the country. I don't think that would happen though if you primarily reside outside of Korea but in the case in question he didn't but was always living in SK and not GB but fled to GB with his UK passport.
These are the area's they are trying to clear up. Most of these scenarios are VERY rare which is why it is taking so long to clear up.
