That might be so for officers, but not for enlisted. The Army writes tech and field manuals at the fourth grade level. The great majority of Army enlistees are not going to be able to read War and Peace. I say that as someone with almost 32 years of military service. But going back to King's statement, it was condescending, elitist and typical of the American hating left. Most people who can't read will either spend most of their lives in prison or they'll be flipping burgers at McDonalds.
As you say, King had a legitimate point to make. Fluency as a reader is essential to a successful career in any field these days. But being a liberal, he took a swipe at the men and women serving in Iraq rather than the illiterates in prison or on welfare.