They are probably worried about accent- many schools in asia want Canadian/Australian/American or British English-First language speakers because they believe that the english learning student will be able to mimic the foreign accent better. They might also be worried about the standard of English you have had in your education- they sadly lump all Asian schools together and deem them not as good as foreign schools- which is definitely not the case!
I would suggest looking into obtaining a TESL [or TESOL= Teaching English as a Second Language] certificate if you do not have one already, or perhaps look in to a course in linguistics- this would throw the accent possibility out.
Another possibility is the schools want a teacher they think speaks like an English-speaker does in terms of slang, contractions and grammatical errors... that might sound odd, but most native English speakers rarely speak "perfect" english, while people who learn English as a second language tend to commit less mistakes, use longer words and speak more formally. Schools now in Asia [I'm speaking mostly about Hong Kong] feel that their own teachers can teach the language mechanics (grammar etc.) but they want a English native speaker to handle the mannerisms part, which is starting to get more valued.
From your writing, I would say you have learned English fairly well, and the language in your question would pretty much match the language used by most people where I live [Vancouver, Canada]- however, there is a large percentage of Asian people here and I am half Chinese myself- it is my caucasian last name from my father that makes me a sought after ESL teacher.
I hope you don't give up! Keep trying- have you actually talked to the principals of the schools in person? That might demonstrate your language skills. Until then, I guess you could tutor
Best wishes!
