They claim the program does produce improvements, but that these seem to go away with time. Clearly there are no differences by 3rd grade.
I didn't look at the claim of early improvements, and it is hard to know what they'd be comparing with, but let's assume that it was true; that Head Start 5-year-olds were better off than identical demographic 5-year-olds without it.
It is possible that these good effects were always going to be transient. I gather that many of the effects of parenting are (extremely bad excluded) generally washed out by the time kids reach their 20's.
It is possible that the good effects were attenuated by clumsy schooling. If the school system is less than ideal for students eligible for Head Start (with and without had the same outcomes), it is not outlandish to worry that traditional schooling might be less than ideal for other classes of students as well.
For example, perhaps boys do less well than girls?
I can think of a lot of things that could be better in school. One is that the kids spend some of the pre-teen time doing non-arbitrary work--paid or volunteer, and preferably not "saving the planet," but rendering services people actually want.
UPDATE: to clarify, I want them interacting with adults and not merely picking up trash or lobbying.