Can School Districts Use Race in Enrollment Policies?

Yes, you can.

The question is - why is Seattle Schools saying, in document after document - you can't?

From the majority opinion:

Importantly, Justice Kennedy drew a distinction between school district plans that rely on the race of individual students and plans that seek to achieve diversity or avoid racial isolation through more generalized race-conscious measures:
If school authorities are concerned that the student-body compositions of certain schools interfere with the objective of offering an equal educational opportunity to all of their students, they are free to devise race-conscious measures to address the problem in a general way and without treating each student in different fashion solely on the basis of a systematic, individual typing by race.
Id. at 788-89 (Kennedy, J., concurring in part and concurring in the judgment).
Justice Kennedy went on to state that race-conscious approaches that do not rely on individual racial classifications are “unlikely” to “demand strict scrutiny” and are likely to pass constitutional muster. In so doing, he also provided some examples of the sorts of approaches that he had in mind:
School boards may pursue the goal of bringing together students of diverse backgrounds and races through other means, including strategic site selection of new schools; drawing attendance zones with general recognition of the demographics of neighborhoods; allocating resources for special programs; recruiting students and faculty in a targeted fashion; and tracking enrollments, performance, and other statistics by race. These mechanisms are race conscious but do not lead to different treatment based on a classification that tells each student he or she is to be defined by race, so it is unlikely any of them would demand strict scrutiny to be found permissible. Executive and legislative branches, which for generations now have considered these types of policies and procedures, should be permitted to employ them with candor and with confidence that a constitutional violation does not occur whenever a decisionmaker considers the impact a given approach might have on students of different races. 

Id. at 789 (citation omitted).
Furthermore, while the Seattle and Louisville school districts failed to show the necessity of classifying individual students by race in their plans, Justice Kennedy refused to rule out approaches that in appropriate circumstances take account of the race of individual students in school assignment.  Id. at 790. He explained that a school district can employ a “more nuanced individual evaluation of school needs and student characteristics that might include race as a component.”  Id. Such an individualized approach would be informed by the narrow tailoring analysis set forth in Grutter,“though of course the criteria relevant to student placement would differ based on the age of the students, the needs of the parents, and the role of the schools.”  Id.

Basically, you cannot use only race for individual students in assignment. But can you use it as part of an overall strategy? It looks like you can.

I understand the gun-shy stance of the district but to say that you can't use race in enrollment strategies isn't quite accurate.