There is another important assumption: what if Monty knows which door has a car in advance and: 1) if you pick the car, Monty opens up other doors as in the re

There is another important assumption: what if Monty knows which door has a car in advance and: 1) if you pick the car, Monty opens up other doors as in the regular problem 2) if you do not pick a car, Monty just opens your door and says "unfortunate pick, contestant". If Monty is an adversarial agent (which is not an unfair assumption in television land), then your strategy changes. So an important assumption to make for the Monty Hall problem to work, is that Monty announced what he will do after you pick your door _before_ you pick your door. Only in that case do you have your counterfactuals correct and in that case it is indeed better to swap doors. It is a hidden assumption in most probability theory type of answers.