You may be right. I'd acknowledge that some of what Heritage produces is methodologically correct. Some questions I have immediately about this piece of work:
1. The data is 6 or so years old. Is current data supporting this trend? How about past? Since they cherry picked two years and provided no additional trend data, the historic and present significance of their data isn't able to be assessed.
2. They make their claims about recruit affluence based on census data. In short, they took the recruits enlistment zip codes and assumed the median income of that zip code was the recruit's family income. This is terrible and lazy methodology. You mean to tell me no poor kids live in say 21401? Annapolis, MD? How about 92105? City Heights near San Diego? Does everyone there make the median household income of $249,000? I doubt it. This method does nothing to correct for the fact that the military may tend to draw in larger numbers, people from households below the median income line of a particular zip code they may be living in. You can't make pronunciations like "U.S. military service disproportionately attracts enlisted personnel and officers who do not come from disadvantaged backgrounds" when your research method isn't refined enough to identify what background the enlistees came from.
3. The study claims that "American soldiers are more educated than their peers" without defining who that peer group is. Apparently, it's every male between 18-24 in the nation because the study then claims it is in comparison to 24% of men 18-24 without a HSD. First, at least a third of your enlistees aren't men. They are women. Where are those statistics? Second, you assume (but again, can't prove, because of your slipshod methodology) that all of these people are socio-economic peers...not valid. Finally, the study neglects to mention that the reason they are so disproportionately educated is that they are required to have a HSD or equivalent prior to enlisting. It's a classic ad hoc ergo propter hoc argument. They look great on paper, but one thing isn't causal to the other.