It's important to note that the usual argument relies on Monty knowingly choosing a door which does not have the prize behind it.
That's true. They did it this way, because that's what Monty Hall really did on the show. He wouldn't have opened the door with the prize behind it, because that gives everything away.
You probably already know that. I'm only pointing this out, because your statement threw me, and I thought you were arguing that the problem was being given wrong.