The primary reason they don't act to commit a coup is because they haven't solved the Byzantine General's Problem.
So you are saying the military does not commit a coup primarily because it's hard to do it? In other words, they would like to do it but they don't because they can't. I have a different opinion explaining why a modern state is able to keep its equilibrium.
... act together to solve it for the protection of the grater good, they would, even if doing so violated the Constitution.
That's where we differ. I think there is no greater good than the Constitution.