I'm exactly the type of person that this study reflects, but I would argue that race is not the significant factor here but the academic performance of the schools. The fact that schools that are largely comprised of migrant communities generally perform poorly is a whole new thread!
The opposite is also true. There was a big fuss in Sydney a few years ago because former students of a selective school (which is the only Public GPS school in Sydney) felt that their children should be given preference over other students. One of their concerns was that the character of the school was being changed because so many of the students came from non-English speaking backgrounds. Most of the best performing schools in the HSC in NSW are public selective schools which have high populations of migrants. One of the only private schools that is regularly placed highly in the HSC results is a Muslim school with a large migrant population. If you take a look at the photo opp with the Premier every year a large percentage of the high achievers are non-Caucasian.