But it doesn't make them beholden to president. And they don't have to follow the president's orders. First, a general will make more money not being a general than being a general. Second, a general can refuse to follow an order if they judge it to be 1) illegal, 2) tactically unsound, ((rather surprisingly long list)).

A president can declare wars, and he can say "go take that country." He doesn't get to micromanage troops. If he doesn't like the way one general is doing it, he can maybe choose another. But a president doesn't get to manage the military. The military manages the military, and the president's control of the military is limited to choosing between options presented to him by the military and between choosing between managers presented to him by the military (the president doesn't appoint generals as he sees fit: he chooses generals that are presented in a list by the military. The military can and has present a list with one name. The president's control of the military is actually very limited.

So no. They are not his. And no reasonable person could see it that way.