But none of that means it's impossible or that it requires luck to break out of (again, assuming they live in a country with abundant economic opportunity). It requires hard work, discipline, and perhaps most importantly, an understanding that the limits their parents ran into that resulted in their impoverished upbringing are not insurmountable.
Part of what allowed me to escape is luck. Yes it required that I developed early and so was ahead of my peers in education. It also required hard work to capitalize on that lead. However, what kept more than one of my cohorts from escaping was bad luck. A family member had a health issue, a legal issue, whatever. Their need to care for their family caused them to miss an opportunity (Get a job rather than go to college).
Unless you're advocating living life as a sociopath and ditching people as 'dead weight', luck is absolutely part of escaping poverty.
In any case, whether you say it is that the parents are poor or that they are drug addicts or don't care or whatever, you're talking about things outside the control of the individual. I cannot choose the parents I was born to.