That seems like an odd complaint - it's not like Koreans don't do that. Unless he's referring to people turning around and opening their own school down the street from their teacher and trying to get a bunch of his students to come with, which I have seen and is a pretty dickish thing to do.
Non-compete clauses aren't entirely unusual. It only makes sense, that if an instructor at a school wants to break away and form his own school, independent of the old school, then he should have the decency to relocate a fair distance away. I certainly concur, that a school owner should guard against potential poaching.
In all of my time in the martial arts, I have seen some highly talented folks get lured away by rival schools, and in turn, open up their own schools just down the road from their old school. Many times, these people would actually be financed by the rival school's chief instructor, since they had intimate knowledge of the old school's operations, and really knew how to hit the old school where it hurt the most.
While most of the students would remain at the old school, there would still be a significant number who would flock to the new school, creating a big headache for everyone involved.
The old school owner loses 20% of his income, and replacing the lost students might not be nearly as feasible, because of the new presence in town.
The new school owner has to deal with the fact that many of the poached students might still be under contract with the old school, and might find themselves short of $$$ if they have to pay for two schools' tuition at the same time. Furthermore, this kind of thing reflects really badly on the new school owner...
Regarding the OP's situation:
Opening up a branch school under the old school's umbrella isn't necessarily a bad idea at all. If anything, you can still use the old school's resources, enlist the old instructor's help at times, etc. This isn't unreasonable, especially since a freshly minted 1st dan is going to need some help in getting his school started. Even if that 1st dan has an excellent working knowledge of school operations, there are going to be many things that he doesn't know about, that he might be unprepared to handle without more experienced help.