Thx for the history lesson re Eton.
Clearly, to all intents and purposes it's a private school as most of its students will have come from upper class and upper middle class backgrounds. The old school tie concept will still hold sway, no doubt! Of course, a few lower class students will study there too on account of having won scholarships/bursaries for having excelled in competitive entrance exams.
Maybe Eton British English School is similar to England's Eton College because it caters for the offspring of rich Chinese people, all be it totally so. I don't think there will be any entrance exams there.