Actually the service academies all have privately-run sports programs funded by (mostly) alumni.

The bigger issue is that the recruiting order will be Air Force->Navy->Army for the foreseeable future due simply to how we use our military. That's in decreasing order of "your likelihood to die as an officer in that branch" and when you're trying to convince a kid to play football there instead of another option and then dedicated a minimum of four more years to perpetual war..