Assuming five races: Given that there are more than five schools, than using a school's past score performance or socioeconomic status is going to be closer to their individual score than the aggregate for their race.

tl;dr School is finer grained than race because there are more schools than races.

That only works when you are able to control for all other factors.

Because school districts in the US are shockingly segregated by both race and income.

A consequence of school funding being based entirely on local property taxes.

And let's face it: also some of our own segregationist instincts. Even liberal San Francisco recently abandoned its eminently egalitarian policy of mixing up the public school population by not basing assignments on neighborhood. I'm not the only progressive who cringes at the thought of not being able to buy my daughter into a particular type of school district.