This is an extremely interesting perspective from the point of view of the psychology of why the problem generates such an emotional response. I entirely agree with you that presenting it as you do emphasizes the combinatorial aspects of the problem and leads to a much clearer exposition and route to the solution.

In truth, this article was meant as an introduction to causal mapping, rather than as an introduction to the Monty Hall problem, so it's natural I should emphasize the fact that, whether its Monty or an algorithm that mediates it, there is a causal relationship between the location of the car and the other door that is opened / the door Monty opens. Seen in the light of your exposition, this is what generates the null cases in your combinatoric table, which reconfigures the probabilities to the answers we thankfully all agree upon.