What do we do when choice conflicts? Say, if a black family wants to send their kid to local school in a predominately white neighborhood, and there is a limited capacity?

It seems like the solution most institutions land on for this kind of thing is just a first come first serve sign up list. Kids who attended the school the previous year, and kids who have siblings currently attending the school, get first priority for open slots. The rest of the slots are first come first serve. This is how my kid's school, a charter school, works.

Sign-up lists self select for parents who are most motivated to get the best school, which has the risk of maintaining roughly positive correlation with the parent's socioeconomic status.

Lottery

My guess is this approach would also be desegregating.

Glad we agree : )