It's not true that "... TEFL jobs [are] given only to native speakers" - I know many Chinese teachers who teach English very well without a trace of "Chinglish" in their accent or approach. In fact, I thought I was speaking to a Londoner on one occasion in China because he spoke and taught with a perfect east London accent! Turns out he was taught by a teacher with that accent and then studied in London for several years.

If you are as good as you say you are, you should have no problem finding a job. I would however take issue with your generalisation that "... a non-native speaker has better ablity... ". This is of course your opinion, and you are entitled to your own beliefs, but your notion that having learned English as a second or other language somehow enables or qualifies you "to connect with the students in a better fashion" is questionable. With that personal experience you may have empathy with students, but empathy and ability to teach are not the same thing. It's also true that learning another language does make a native speaker more aware of grammatical constructions and differences.

Your question should really be addressed to the those who do the recruiting. Have you applied and then been turned down? With your qualifications and experience perhaps you should re-think what you write on your resume and how you market yourself? Incidentally you might also consider what you write because there are those who would frown on poor grammar and 'typographical' errors. Good luck!