It's often suggested that Koreans in Japan are somehow deprived of Japanese citizenship. This is not true. Rather they are not allowed to naturalize in Japan while retaining their Korean citizenship (yes, I realize that is not the whole story and there is a lot of history there - nevertheless, there is nothing stopping a Korean who has lived in Japan long enough, or was born there, from acquiring Japanese citizenship if they choose). Many countries have this sort of policy yet somehow it's controversial when Japan does it.
That said, mass immigration is unlikely, as you say.
It is the same for the US. People that naturalize are "required" to sever citizenship they hold with other nations. However, many people fail to follow through and notify previous host nations.
That said, mass immigration is unlikely, as you say.