Yes, you're right, it's totally unfair to non native speakers, as we've said in the many posts here, but you have to look at it from the employers point of view really. If you're not a native speaker, how can you know the nuances of the language, the "ins and outs" of the language etc. Only native speakers can know these things as it's their language and they've lived with it their whole lives.
There are many native speakers who aren't white too, like the African Americans and others but they're discriminated against a bit here too. White skin and a passport from a native English speaking country: U.S, U.K, Canada, Australia, N.Z and maybe, South Africa and Ireland are the ones who are more readily considered able to speak, therefore teach, English better than a non native speaker. A neutral accent helps too, but who among us can truly say we have one of those as we all have the accent of the country we're from.
As Margy has already said previously in this forum, maybe you should consider finding a job other than teaching English abroad, but don't let this stop you from trying. There are places in the provinces and more remote areas who'll accept people who aren't native speakers. I worked in Hunan for a term and there was an African woman from Cameroon who taught English there. The trouble is, that a lot of students couldn't understand her because her accent was a mixture of her own language and French and when she was confronted about this, she got angry and was rude to the students. The school kept her on though, because they couldn't get native speakers to go there because the college was in a little, poor ass, provincial city that was really badly underdeveloped and paid a tiny salary. Living conditions were terrible, but I kinda liked it there, I didn't mind staying as long is was only for a term. Maybe try applying to the cities like that.....Just and idea for you. Anyways, don't be discouraged too much, keep trying and remember that what you put "out there", you get back. Good luck!
just a few cents folks,
