This question has been the object of heated debates among citizens of countries where English is widely spoken as a first language but which are not qualified to teach English in Korea, i.e., India, the Philippines and several countries in African and the Caribbean. Some have accused Koreans of being racist. I have heard the craziest excuses regarding this question. Some (Americans) even told me that African Americans, Indians etc. were not genetically feat to teach English. Others claimed that Black people and Indians had an "accent". What would a linguist think?
If you ask me, the fact that they hire teachers from Ireland or South Africa, Scotland or Wales, Australia or New Zealand means Koreans have no problems with accent. You can argue all you want, a Scottish accent, a Wales accent or a South African accent has as many differences as the differences between an American accent and an Indian accent. To be more specific, if you record a Scottish person, an Indian person, a Philippino, a Nigerian, a Jamaican, a New Yorker and a Texan pronounce the same word, the charts analyzing tone, stress, intonation etc. will all be completely different.
So why White people. Koreans do not consider accent as a requirement to teach, they consider socio-economic factors.
Koreans want English teachers to be temporary teachers and to stay in Korea temporarily. They want them to be people who, once they go back home, can find better jobs and make more money. They don't want teachers who teach for the money, they want teachers who teach because they want an adventure, they want to be in contact with Korean people and who are happy to teach.
English teachers from African and Asian, or English speaking countries in the Caribbean would come teach English in Korea because of the high salary, not because they want to teach English. Most importantly, they would be reluctant to go back to their country and would try every trick in the book to spend as much time as they can in Korea. They would also bring over their families, and settle in Korea, and since their stay in Korea will be motivated by economic rather than cultural motives, they will have very little interaction with Koreans. This is not something the Korean government would not like to happen.
Finally, Koreans learn English because they have to in order to get a job. Their role models are successful businessmen. As there are very few non-White western successful businessmen, they want their teachers to look like Bill Gates and Steve Jobs. Therefore, even Moroccan, Iranian or Turkish teachers who have more or less the same features as Bill Gates and Steve Jobs (Jobs is part Syrian) they will be hired by hagwons to teach English even though it is technically illegal.
Therefore, Koreans want their English teachers to:
1- Leave Korea as quickly as possible
2- Look like middle class western citizens.