Since your kids attend private schools (as do mine) I have no choice but to assume that you have no idea what hell holes many public schools have become.
Assigning seats at lunch is a reasonable practice, hardly worth the "Rosa Parks, Martin Luther King, Tyndale burn-me-at-the-stake-for-standing-up-against-fascism-and-for-freedom" perspective.
It isn't asking much for the little darlings to sit in an assigned seat for a 30 minute lunch break.
I don't believe in punishing the many for the crimes of the few. But hey, that's just me.