Well, this study is hopelessly confounded by lack of controls. Although present economic circumstances were putatively controlled, children going into Head Start tend to be more from one-parent families, parent/parents less educated, fewer good older role models and all the rest. So what you say may be correct.
Yes. Head Start was designed specifically for poor, black, children of single-parents [though I'm sure that wasn't in the brochure]. They would benefit the most from a program like this.
That they don't benefit wouldn't appear to be any indication that children from better educated families would, either. In reality, you don't need a study for things like this; but I do realize that politics is what it is.