I didn't model it that way, but who is to say (given the problem statement) that Monty wouldn't choose to open your door first if you didn't choose the car? In that case the analysis comes out to what I showed.

Indeed in the slight variation of this game known as the shell game, it is indeed traditional to show people right off the bat that they got it wrong. (It is also traditional to remove/re-add the pea by sleight of hand so that the chump playing always loses..but let's ignore that fact.) Indeed that precedent suggests that there might indeed be room for a game show where the host tries to make life hard for the players. Indeed the existence of reality shows suggests that the traditional game show format is not the only possible one with an audience.

Furthermore even if you do (based on cultural knowledge) find only one possible assumption natural, it is still good to analyze the problem clearly enough to know that that is an assumption. Because while implicit assumptions often are right, every so often you'll run across a problem that you can only get straight by being able to find and correct a wrong assumption that you made. (In my experience, "every so often" comes by pretty frequently...)