The biggest factor that determines school success is motivation. If a student is motivated to stay in a superior school, then the chances are that they will be able to do so. A hard working student isn't likely to fall behind in the first place.
Sure, children of poverty have a strike against them from the outset. The current system places children of poverty in the same school, where they have a second strike against them in attending a school with low standards.
Motivation is key, but I am also getting the sense you also understand there is a nuance at work here. Many poor families, for instance, those that I work with, can be quite motivated to succeed. The problem is when life intervenes, and it's quite likely it does. We have parents that are chronically ill, some die, kids get thrown into one extreme internal or external situation after another.
I've seen kids with more "grit" than most of the population, but for one reason or another, find themselves on the easy track to failure or dropping out. Then, on the side, professionals can fail these kids too by getting angry at the kid or the parent without knowing (or caring) what is going on for that family. I've seen that happen, and sadly, we've lost parents. Then, naturally, the kids aren't doing well because they lost their family members and natural supports. Any number of social maladies can intervene after that, including behavioral issues and, if that kid is labeled just right, placed in institutional facilities--on the quick path to incarceration if not a life-long resident of a non-criminal facility.
Just consider what it's like when you are the kid of a single parent and that parent first is chronically Ill, and then later, dies. The kid takes on a lot more responsibilities to help that parent and that household, putting their studies behind. Then the parent dies.
Others I know, they themselves are chronically ill, almost in the hospital every week.
I like where the personal responsibility and good parenting argument goes, but I think too often it is seen through the perspective of families that don't have to deal with strictly economic issues, chronic illness and dependency, mental illness, or other outside forces.
This reliance on the market, "good parenting," or grit needs a lot more cultural competency.