I think you could probably develop that idea a bit more from simply watching to actually bringing people together. I have found computer games like Arma to be a great medium for community building. Its probably the only game that you can decently recreate a "military" environment in, where what you do as a unit is really only limited by the imagination and creativity of the mission maker. For the majority of the rest of the week when the group is not together as a whole, members of the group will bullshit around on teamspeak and play whatever other games come to mind.
If you want to stream, you need content. This would offer both content to be viewed and a gaming community to bring vets together. If carried out with even a modicum of competence, the combo of community and content would probably become a self licking ice cream cone and take a life of its own.