AxisMundi
E Pluribus Unum!!!
At the k-12 level, 'good-school' is a misnomer. That's because in those levels, the school admissions are based on geographic proximity to the specific schools. And since 'good-schools' are more likely to be in affluent and wealthy neighborhoods with more property tax money and funding to perpetuate its 'good' status, the virtue of you being able to send your kid to a 'good-school' most likely means you yourself are likely to be relatively middle-class, affluent, or even wealthy. And it's those factors that contributes to a child's likelihood to do better. Although, as you've mentioned, it's not a guarantee. It only increases the likelihood.
A parent can request a specific school for their child within any city school system.
It might take a few years if there is already a alrge student body within that school and/or a long waiting list, but it will happen eventually.