I know a lot of this debate turns around what is classified as a sports expense. For example, I believe WP referenced NAPs as a hidden expense for the athletic program, and there's an argument to be made for that, given how much NAPS (and USMAPs) are used to redshirt for athletics. However, even under that argument, I think one would have to at least seriously consider that NAPS would still exist even without its relationship to athletics, as it advances two additional missions: facilitating the entry of enlisted personnel to USNA, and advancing diversity at USNA. From anecdotal evidence, I would surmise that the NAPS students who fit into these latter two categories are outweighed by those who are recruited athletes (and some of the athletes also fit within the diversity category, of course), but even taking a very broad view of athletics-related expenses I do not think one could appropriately chalk up the entire operating budget of NAPS to athletics.
Another point, and this is not a new one I know, is the unquantified value of a three-hour commercial for Annapolis and West Point each year in the form of the Army-Navy game. And the networks pay Army and Navy for the right to air said commercial! The fact that the commercial is in the form of a sporting event, and thus reaches all the eyeballs of people who fast forward through "ordinary" commercials on their DVR/TIVO, is an added benefit. And sports-loving alums donate -- much of that money returns straight to the football fold, but some of it does go to support non-revenue sports.
I have no doubt that WP is correct that looking at a budget of the NAAA (and the West Point equivalent) may well leave an educated observer with more questions than answers, and I know of a few issues myself (making varsity assistant coaches "instant ensigns" and justifying it because they help teach PE to non-varsity athletes). However, in large part because of the Army-Navy game, the football program does at least bring IN a significant amount of cash, even if we can't agree on how much cash is going out.
Moreover, for good or ill (and maybe largely for ill), American colleges and universities have big athletic programs, revenue and non-revenue sports alike, and for many American high-schoolers, getting to play or watch sports is a big part of the college experience. I expect the academies feel that, were they to go away from athletics (dialing down to Division III is an interesting, and different, discussion) they would lose many highly qualified applicants. And last of all (I promise), there's always the "I need an officer for a secret and dangerous mission -- get me a West Point football player" rationale: in a physical and demanding end job, varsity athletes may bring some highly desirable attributes.
Apologies for the lengthy post, and thanks for an interesting and thought-provoking debate.