The president's control over the military fits with the general separation of powers. Congress declares war (ha! in theory, anyway), and the president executes the war by directing the military.
Alexander Hamilton in Federalist Paper no. 74 basically says the choice is so obvious as to require no explanation:
"THE President of the United States is to be "commander-in-chief of the army and navy of the United States, and of the militia of the several States WHEN CALLED INTO THE ACTUAL SERVICE of the United States.'' The propriety of this provision is so evident in itself, and it is, at the same time, so consonant to the precedents of the State constitutions in general, that little need be said to explain or enforce it. Even those of them which have, in other respects, coupled the chief magistrate with a council, have for the most part concentrated the military authority in him alone. Of all the cares or concerns of government, the direction of war most peculiarly demands those qualities which distinguish the exercise of power by a single hand. The direction of war implies the direction of the common strength; and the power of directing and employing the common strength, forms a usual and essential part in the definition of the executive authority."
If anything, the more important characteristic of the military is that it's under civilian control to begin with, in contrast to many countries where the military is independently governed (which leads to *all sorts* of trouble).