[tel]

You're absolutely right and my argument fails, but it doesn't invalidate the point I wanted to get at. In the long run, your coin flipping method does cause the game to behave as though there were only two doors, but in any given game you'll know whether or not the host is going to choose a door.

That means that, conditional on a good flip, you may still be able to act to your advantage since there's information transfer from the host to you.