I was going to throw Colgate and Fordham into that group too.
Army / Navy sort of fit the cultural mold, except they both insist on an FBS football schedule, to be used as a recruiting tool.
Navy seems to make that work (sort of mediocre in football, but playing some teams that draw crowds.)
Army football has been way over their heads in FBS for the last few decades, and seems to schedule a lot of MAC schools who wont embarrass them so much.
If they really can't elevate their game above the FCS level, I'd ratehr see Army just play FB vs. the Patriot / Ivy League schools.
Colgate and Fordham are solid options.
Army just has been what Navy and Air Force have been. Since winning the Commander-in-Chief's Trophy in 1988, Army has only won it once (1996). Since the 1996 season, Army has only won 4 games combined against Air Force and Navy (2 times each) with their last victory against Navy in 2001. It's sad, but it's reality.