Sure it's only one branch of the probability tree of the game as a whole. But the way the problem is frequently framed is this (I acknowledge that the OP does

Sure it's only one branch of the probability tree of the game as a whole. But the way the problem is frequently framed is this (I acknowledge that the OP does briefly make the important distinction though): "The game has three doors, two with goats and one with a car. You pick one door. The host selects another for free and shows you a goat behind it. Do you want to switch doors?" My point is that, given this scenario, to make an informed decision you need to ask the host: "Did you know where the goats and cars were? And did you open the goat with certainty, or did you do so at random?" The hosts answer to this question then determines whether you should switch to gain a 33 percent advantage or that it does not matter.