The only explanation that convinced me that my original guess was incorrect was this one (which is alluded to in the article):
If there are three doors, the odds that the car is behind any of them is 1 in 3, or 1/3. Once you select a door, the odds are 1/3 for your door, 1/3 for each of the other, unselected doors. Put more simply, 1/3 for your door, or 2/3 that it is not in your door. Once another door is opened that does not have the car, then the odds of it being your door are 1/3, and the odds of it not being your door are still 2/3, so it makes sense to switch.
Thank you. Adding the temporal domain and system states is what makes this easy to understand.
At the time of selection you freeze the probability of the sets (as this is the macroscopic world and the implications of the state in the other rooms must have collapsed due to interactions with all sorts of radiation from the rest of the universe).
If there are three doors, the odds that the car is behind any of them is 1 in 3, or 1/3. Once you select a door, the odds are 1/3 for your door, 1/3 for each of the other, unselected doors. Put more simply, 1/3 for your door, or 2/3 that it is not in your door. Once another door is opened that does not have the car, then the odds of it being your door are 1/3, and the odds of it not being your door are still 2/3, so it makes sense to switch.