The Chairman of the Joint Chiefs of Staff is, by U.S. law, the highest-ranking member of the US military. The relationship of presidents to their military is no different than that of a mayors/governors to their police departments: While they are in command of their military or police - the latter are not serving in the police and former are not serving in the military. But according to the Constitution, the president is the commander-in-chief, and he has the ultimate authority over the armed forces. The National Museum of the Pacific War in Fredericksburg, Texas honors all of the presidents who served in the war and includes Truman and Roosevelt.
This thread was initially discussing "military service." That's what my posts were addressing.
But speaking only of "service," State Department employees abroad are called the "US Foreign Service." And while the President and many other civilians in the federal government (State, CIA, FBI etc) may be honored for their service to their country during a war, their "service" is not "military service," no matter how critical it was to the military effort or even if their title is "Commander and Chief."
Unlike many other country, the US has for its entire history held firmly to its doctrine of Civilian Control of the Military - which could not be the case if the President was not a civilian but part of the military:
