Have you ever heard of 'regression towards the mean'? On average, exceptional parents tend to have slightly less exceptional children.
In any case, there's no need to sit here and think up a priori explanations for this correlation. I'm afraid I don't have time to look them up right now, but I would guess that there have been studies done that examined the performance of adopted children of wealthy families, and I strongly suspect that these studies show that the adopted children perform comparably to biological children of wealthy families.
These hypothetical studies would not contradict an explanation which emphasized upbringing, rather than income or genetics as a determinant of success. Parents bring their time and perspective, as well as their money, to the table.
For such an explanation, I'll note that as the article mentions re. savings plans, much of the behavior which perpetuates poverty tends also to be associated with a short time horizon, or a high discounting of the value of future income. It may be that these practices are largely adopted for lack of educating and training the child of the alternative: that sacrifice and current effort can lead to greater happiness over time. Parents would be well-situated to impart such a lesson.
But my explanation need not be the only factor at play, it only serves to show that the view that income begets income (and thus needs to be equalized) is not the only conclusion supported by the facts.
In any case, there's no need to sit here and think up a priori explanations for this correlation. I'm afraid I don't have time to look them up right now, but I would guess that there have been studies done that examined the performance of adopted children of wealthy families, and I strongly suspect that these studies show that the adopted children perform comparably to biological children of wealthy families.