Why is it that the military must obey their boss, the Commander-in-Chief, but yet he doesn’t have to obey his boss — the American people, an overwhelming percentage of which don’t want to do this?
It's a systems problem, or alternately, a command and control problem. The military is supposed to obey the chain of command in real time. The President only faces the people every four years, and their decisiveness is diluted by party factionalism, multiple issues, and propaganda.
In parliamentary systems, they trade the possibility of more rapid replacement through no-confidence votes, against the possibility of indecisiveness and paralysis due to changing leadership.
One of the most striking things in France in the 30's was that the political factions were almost evenly split between republicans, fascists, and communists. Governments rotated through every few months, and the indecisiveness showed when the Germans invaded in 1940.