In a few different sites and blogs, there's a lot of mention of military presses being bad for the shoulder. Are they any worse than any other exercise? and if so, what would be an alternative method to get the same results?
From what I've read, the problem isn't the motion as much as the bar position. In short, with the bar behind the neck, it is very easy to injure yourself, and in people who lack the proper shoulder flexibility to cause rotator cuff injuries.
The "press" in its general definition of a standing lift with the bar in front of the head is a very safe lift and does not come with the extra stress on the tendons and ligaments. It's also very beneficial at addressing all the muscle groups referenced in your diagram.