I mostly agree with you, but a couple minor caveats. First of all this is a relatively new thing, basically starting with Vietnam. Secondly, the children of elites are strongly expected to go to college immediately after high school. The military is generally  set up either for non-college grad enlistees, or college grad officers. In the old big expansions of WWI and WWII college grad draftees and volunteers were in fact generally made officers. There's two ways to become an officer - the service academies or ROTC. They army has cut down on ROTC programs in Blue areas and in top private universities and colleges, while expanding them in the big public universities  of the Red states. That means that you often need to make significantly more effort to be in the ROTC if you go to the sort of place where the children of the elite disproportionately end up. Thus in NYC there are no ROTC programs in Manhattan or Brooklyn, nor are there any at any of the CUNY campuses. The only ones are one up in the Bronx (Fordham) and one in Eastern Queens (St. Johns). If you're an NYU or Columbia student, enjoy the over one hour each way commute to every ROTC class. Similar problems for three of the four top CUNY campuses (Brooklyn College plus Hunter and CCNY in Manhattan, Queens College isn't that far from St. Johns)

But like I said, I think you are mostly right on this.