Steep to me is where you don't have enough forward speed to fly away,
You can fly a climbing approach and still have enough power to fly away providing the power margin is good enough (good forward speed or updraughting air). Obviously the speed will have to decrease at some point but not until you are assured of making the LS.
I don't know about Army training but the RAF also teach the steep approach on the basic mountain sortie. There are times when this is appropriate. After the 'basic' sortie student are then taught advanced mountains which involves staying on the updraughting side wherever possible (you then don't need to worry about the nast demarcation line) and flying level (and climbing approaches).
One of the problems a discussion like this (rather like the steep/shallow comment) is that everyone has their own picture of a typical mountain approach but in reality they are all different. Are we talking pinnacle, ridge, valley , bowl, etc? What wind strength are we assuming? How much spare 'power' does your machine have? What is the weather? All these variables need to be taken into account. There is no one answer.