The reason that these numbers fascinate me so much is that numbers tell the story of the aggregations of people. Individual Horatio Alger stories are inspiring, and I never want to belittle their importance, but these stories are remarkable precisely because they are so rare.
But why are they so rare? You're about to get the three point version of the thesis that never was, so get excited.
Poverty forces too many decisions
Consistently making good decisions requires systems more than it requires will power or self control (although more of either never hurt). Creating effective systems requires access to resources which is something that most people in poverty don't have. So the systems born out of poverty break easily and force more decisions.
The problem isn't just the decision fatigue, its that the decision making starts at too young of an age. Children in poverty learn to ration their requests at a young age, which means that many kids in poverty spend a lot of time wishing, dreaming and deciding whether or not to ask for what they want. My armchair economist/sociologist theory is that by the time kids start making meaningful life decisions (regarding work, lifestyle, sex, and education) they are already deeply entrenched in the threatening pattern of decision fatigue.
Systematic risk introduced to the budget
In software engineering, a system is a group of components that work together to achieve some objective (in my world, the objective is usually to quantify something). The system "risk" is the probability that any one component fails, and the objective cannot be achieved. Usually software engineers will mitigate risk by either reducing components, reducing complex dependencies, or building routines that will systematically fix known failure points.
On the poverty level, the components, the dependencies and the solutions are often outside of your control. Imagine briefly what can go wrong in between the time you wake up and the time you go to work, if you're on the poverty level. Your mom fails to show up at your place to watch the kids in the morning, so you've got to get them to the bus yourself. Your phone is out of minutes, so you can't call in until you spot a neighbor in their yard. You're out of money and out of gas. You were going to take the bus, but with staying with the kids you're already 45 minutes late. You run out of gas a mile from work, pull over to the side of the road and begin to walk the rest of the way to work. You debate whether or not you should call a tow truck, and you're manager tells you that this if you're two hours late to work again you might as well not come in.
The point of the budget at low income levels isn't that it will turn your life around. However, the regular practice of budgeting could set you up to take advantage of opportunities that arise in the future. Perhaps that stroke of good luck will hold out for a year or more. Perhaps your tax refund will be unusually large, or you'll finally find a job where you earn a living wage. The last thing you want to do when opportunity finally presents itself is to rely on your limited willpower to take care of your finances. Having a budgeting system in place will allow you to use your limited decision making power to take advantage of your opportunity rather than wasting the decision making power on basic financial decisions.
Higher income and reliable access to necessary resources (childcare and transit top my list) are one set of prerequisites to enabling class mobility, and the ability to make reliable personal systems is the second requirement. Although a budget alone won't solve the problems of poverty, it is still necessary for class mobility.
This leads me to one last question: If you've got the habits in place, what is the likelihood that you will advance economically? My opinion is that if you are earning above the lowest income quarter (approximately $25K annually), then personal finance can help you advance. At $25K per year, your money (in combination with government subsidized childcare) can buy you stable access to resources that will allow you to advance if you're willing to try (probably a few times). You are in a position where the budget and the hustle will pay off. Below that level, I think you'll need great luck or systematic intervention. What do you think?