I've been reading a bit more about David E. Kirkland and came across an interesting article featuring an interview with Kirkland in Jan 2017:
Originally Posted by interview article
There actually are many diverse neighborhoods, and — at the middle and high school level — we have lots of school choice. What will the center’s role be in trying to harness that? Are there opportunities there?

New York city is this really interesting place where young people don’t necessarily get out of their neighborhoods. They may have a choice, but they don’t necessarily feel it. It’s the caged-bird effect. You can open the door to the cage and say ‘Hey, there’s a world out there, bird.’ But if that bird has been behind the cage forever, the bird is not going anywhere because its mind is caged.

People don’t perceive the choice, if there is a choice. And that perception is necessary in order for there to be a choice.

I believe Kirkland's stated concern may be well-answered by suggestions from the Policy Brief in the OP:
Originally Posted by excerpts from Policy Brief report
- Perhaps, all students who reach a certain threshold on their state ELA and math scores, for instance, could receive an invitation (or automatically be signed up) to take the SHSAT.
...
- Schools or community-based organizations might be able to improve access for disadvantaged students by offering free, high-quality SHSAT preparation.
...
- Providing families with more information about the specialized schools, earlier on, might help seed interest in attending.
To differing degrees, each of these proposals raises awareness and enhances the element of choice.