prout,
On many levels, I'm sorry for your DS's disappointment. I can certainly understand your desire to "get to the bottom this."
I am struck most by cavalier vibe and assurances given your DS, knowing that his SAT scores were so low--yes they are low--and knowing how competitive the Academies are. Someone should have flagged the disconnect between and 1130 SAT and a 4.0 GPA. Even more, there should have been a tip off, like low PSAT or earlier SAT sittings. As someone stated earlier, those spots in the lowest 25 percentile are reserved for special cases and I won't speculate on what constitutes a special case other than to say that recruited athlete would fall in that category.
If DS's desire is to try again next year, the standardized tests (ACT or SAT), treated a separate school subject altogether.
Both my DS's did well on the Math SAT, but had problems with Verbal. We used a private tutor for each, for 4-6 hours. He was a law student. BTW, we didn't pay the tutor to watch them take the tests. He taught them the techniques which they employed while taking the practice tests at home. He then analyzed the results with them looking for consistent weaknesses, most of which were correctable.
In both sons' cases, it was intimidation by and dislike of the Verbal section feeding off of each other. Once they learned the "tricks", it became more of a game. There are general strategies for test taking, but also tactics for standardized tests. The SAT and ACT even have their own individual peculiarities. There is a specific way to read each question. There are tricks and tactics which allow you to eliminate answers almost without reading the question. The tactics are few and they are simple. Learn the skills and then employ them with practice tests. After doing that, he will walk into the test looser and with greater confidence.
Whatever he does, don't simply grind through endless practice tests. He'll only frustrate himself. The best athletes arrive at the game with confidence based on an attention to detail during practices which he/she made meaningful.
I have no doubt that if your DS has a 4.0 and AP classes in Math, Science and Lang/Lit under his belt, he can raise his scores considerably.