You're 100% right in the legal sense about Korean citizenship and yes it makes perfect sense. I don't think that was in doubt anyway. At the end of the day though if one feels they're Korean, it doesn't require validation.
It's not about being accepted. If family ties and ancestral roots are strong enough for one to claim that he or she is Korean, surely there should be an effort to integrate into the Korean culture. This effort, unfortunately, is non-existent in K-Town. It's got nothing to do with listening to SNSD, although I do see plenty of Komericans going apeshite over them anyway.