The military does support competitive, interservice sports teams. Examples: http://www.usafsports.com/AllAFSports.htm
Now selling paintball, that's a pretty tall order -
Possible positive - The network of MWR paintball programs.
Negative - The Air Force is cutting people, selling a recruiting/ad tool is a little redundant.
Academy programs are personally funded with support from local fields, teams, and associations of graduates. Regulations prevent solicitation of sponsors (by the federal employees). Just an FYI AFA paintball's been around a while and Navy's program has been playing NCPA even longer. Army might be even older....
Edit - And just getting those three programs to the same venue for an exhibition has been impossible due to training requirements. I can't imagine fielding an operational team with how hard everyone is working these days.