Your "rather than a shared public school experience which enables societal cohesion" is what scares the natives, dontcha know.
Whose ethnicity determines what's "shared" again? Those people who believe in lockers but no personal workspace, no cubicles?
In a real public school everyone gets at least as much personal space as a cube farmer gets in an office tower. That's a minimum student demand we'd hope to meet. A private study space, yours for the year, but in a shared space (yes, an office tower most likely -- what you never went to school above the 10th floor?).
What we do not have, have never had, is one victorious ethnicity before which all others must bow down.
What we do have, or try to have, is a way of managing dynamism as the melting pot roils but never settles down into just the one "gray goo" the lazier social theorists may yearn for, as it'd make social theory so much easier.
I find it funny how STEM-starved Americans feel insulted and somewhat dazed that Sufi inspired teachers from afar would seek to spread excitement about math and science. You'd think a nation of missionaries would get that.
Lesley Stahl on 60 Minutes that time: why would Turkish people have anything to teach *us*? (her attitude -- she called it counter-intuitive). Many Americans have no clue how Middle Eastern cultures kept the ancient Greek civilization alive while the Latin cultures dove into darkness. They think Al Gore invented the algorithm, not Al Khwarizmi.
Why not take the same attitudes we have towards dining out, when variety is much appreciated. You're this family, looking at schools. Do you want the Greeks (pretty orthodox), the Geeks (more like code school), the Russian-flavored (lotsa lit), or Burger King (all American party lovers)?
Every neighborhood has its diverse offerings, its flavors (always shifting), presumably somewhat reflective of the people who actually live there. This is not a top down imposition.
People of Turkish heritage are all over the place (as we're finding out in the news these days) and many have American citizenship. However if more franchises spin up like Harmony and those, charter or not charter I don't know, marketing as Singapore-tied, or Czech Republic, am I gonna complain? Off the bat, no. I'd like to sample, see the Youtubes. Diversity in itself is not a problem.
It's not about little enclaves all learning to hate each other. It's about cross-enrolling across neighborhoods, on purpose picking whatever magnet or immersion experience.
Where but in America do families have such a smorgasbord of offerings and opportunities to synergize? It's what made America great.
Why not celebrate our heritage rather than insist on dumbing it down and enforcing some uniform pabulum, a lot of it recycled from the UK, from which independence was won you might recall. Or go with a Pearson prime UK-template flavor, fine, if that's in your taste buds. Free country. As long as there's choice (staying home one of them).