(-red-faced former enlisted guy)
It's a common misunderstanding. Especially of junior enlisted types. By the time *most* folks go to NCO leadership training, they know the officers put their pants on one leg at a time, but are still horses of a somewhat different color, with different rules and different responsibilities.
I think the the folks who wrote the enlisted oath, especially the 1962 version, never thought there would be a conflict between supporting and defending the Constitution and obeying the orders of the President. And of course there is no conflict if the man in the Oval Office truly is the President. (as long as the orders themselves are lawful).