I think the intention of the law against foreign teachers doing private lessons is to keep people from being in the country illegally.No, the restrictions against private tutoring have everything to do with trying to maintain an egalitarian playing field in a country that has that as a goal but which is forever realizing it is nowhere near that.
Private tutoring has long been seen as a way for the rich to maintain an edge over everyone else in the hyper-competitive race to get into top-tier Korean universities. By making it illegal, they have tried to level the playing field. The illegality of private tutoring has been tried in varying degrees — with most Korean nationals also being prevented from doing it at different times. Only recently have some of the restrictions been loosened, with college students, for example, being able to teach privately (if they report it) but not teachers and certain other individuals in a position of authority.
Though the laws also apply to different foreign nationals in different ways (depending on visa), the laws themselves were created for an entirely different purpose.