I live in a city-adjacent suburb in Chicago, Oak Park, directly butting up against Austin, the largest high-crime overwhelmingly black neighborhood in the city of Chicago.
Oak Park is tiny and known for its school system.
The western side Austin Ave, which divides Chicago from Oak Park, hosts an unbroken row of multi-family dwellings, many of them large apartment buildings. The occupants of those buildings are people who would otherwise live in Austin, but move to Oak Park to get their kids into Oak Park schools.
This has been going on for decades in Oak Park, and we continue to have excellent schools, and continue to be a high-status magnet for professionals who want to live somewhere connected to the CTA trains but not part of CPS schools.
It doesn't happen in Kennilworth or Wilmette, because those suburbs aren't directly adjacent to troubled neighborhoods in Chicago. But there is absolutely no reason to believe that the carrying capacity for lower-income students of those suburbs --- the majority of Chicago suburbs, which like Wilmette and unlike Oak Park, aren't city-adjacent --- is any less than that of Oak Park.
So, no, I don't think your hypothesis about inner-city kids disrupting suburban schools is accurate, nor do I think "instant white flight" is going to happen.
"White flight" was an irrational, racist reaction to housing integration. It's not a law of nature; it was/is a growing pain of the US getting over centuries of segregation.