I always saw starfleet then more than just a military force. I think that was what Riker meant when he said that a military organization. Kind of separating them from say the Klingons and Romulans that do have full time military organizations.
I do see starfleet as an evolution of the US Navy now, multipurpose. Able to defend the Federation but also provide exploration and humanitarian missions. On the ship, it would be a matter of commander's choice and style of leadership. Kirk is more direct and in an earlier time. He is still somewhat democratic and values Spock and Bone's info. Picard is a bit more democratic but there is less threats, at first, in the 24th century. Picard did have final say and could be very direct when he wanted to be. And as far as being a solider, I would not sleep on Picard, he proved he could throw down when needed.