An Aussie rappel is useful for when you need to see and work going forward. It's particularly useful on steep slopes, not just down vertical or extreme slopes you are going to run down. You may need to clear brush down a steep slope, working in front of you as you go, something you obviously can't do if you aren't forward facing and able to stand safely.
You may have to rescue someone who has fallen down a steep slope, and the situation dictates you either cannot or do not want to go past them or get behind them. Imagine someone has slid down a steep incline to a cliff. It may be easier to work in front of/above them than off to one side or below them.
Keep in mind, when training to Aussie, you may have someone on belay, but in practice, it's typically used in situations where you aren't going to have anyone on belay. It requires practice and technical proficiency. There's nothing that says you have to do it with one hand free, either. In fact, you normally have your non-braking hand on the line behind you--unless the situation specifically requires you to have one hand free. That usually means it should have something in it, or you are engaged in an open air rappel and need it for balance. As they instructed us at the Army's Air Assault School, if it comes free behind you, you leave it free because there's always a chance that re-stressing the line while grabbing it, you could cause an accidental disconnect.
It's non-military applications are special instances where you might need to peer over edges or overhangs, do inspections on steep grades/inclines and such. An example of an application that shares across military, law enforcement, and civilian use is the overhang/edge scenario. You can go to an edge, lean forward and out over it, lock yourself in with a forward incline, allowing you to see down a vertical face like a building, cliff, or extreme slope...and from that vantage point provide direction to others below who are engaged in a traditional descent. Or communicating with someone awaiting rescue below, providing direction and encouragement.
Yes, it is mainly used in military and law enforcement applications where you must enter the immediate vicinity of non-friendlies or are at risk of immediate engagement. Nobody wants to get shot, and you want to get shot in the back even less. And you want to be able to shoot back. It's a warrior thing.