No, that is not how the original Monty Hall problem was formulated. The original said: "the host, who knows what's behind the doors, opens another door, which

No, that is not how the original Monty Hall problem was formulated. The original said: "the host, who knows what's behind the doors, opens another door, which has a goat." Your formulation of the problem changes it from the original Monty Hall problem to something more similar to the Tuesday Boy problem, where the trick is that you have pre-selected a bunch of outcomes and discarded others, changing the probabilities. And going back to the comments above this, it's not the case that Monty choosing a door by random is the thing that matters, it's the thing that in this version he chose a door by random and you discarded all the outcomes where Monty chose the car. It's not the case that people are confused by the problem because they interpret the problem like you did. People understand that a goat door is always eliminated as part of the game, and they still don't agree that switching is better.