I agree when you say “stay,” but we have “stayed” in Germany for 70 years. We would have had some bases there, and indeed SHOULD have taken some of their oil for payment in a “treaty.”
I’m still mystified, as a historian, that we are apparently the only nation on earth to win a war (Mexican War) and PAY the loser.
the best examples are Korea and Vietnam. Like Iraq, its neighbors and enemies were intent on taking over once the US left. With the former, we stayed, with the latter, we gave away a fragile stability due to nasty domestic politics and cost. In turn, one was a success, the other not.