What Navy has done in part to be dominant in shot is they prioritize winning football games. They put money into hiring top notch coaches. The Naval academy coach’s salary is more than double the USMA coach’s. They also have a number of players who went to the NFL either abandoning their service committments entirely or getting a specious waiver to serve in the reserves instead of active duty.
Is that the role of a service academy? I don’t know. Are they drawing better future marine and navy officer candidates by having a good football team? Maybe. Probably they are getting some cadet prospects favoring Navy over Army because of their football team, but finding people to fill the seats in the incoming classes with high quality cadets/midshipmen hasn’t been a problem.
If any academy had an affinity to athletics by nature, military ones would.
https://en.wikipedia.org/wiki/Roger_Staubach
Roger Thomas Staubach (born February 5, 1942), nicknamed Captain America and Captain Comeback,[1] is a former American football quarterback in the National Football League (NFL).
He attended the U.S. Naval Academy where he won the 1963 Heisman Trophy, and after graduation he served in the U.S. Navy, including a tour of duty in Vietnam. Staubach joined the Dallas Cowboys in 1969 and played with the club during all 11 seasons of his career.