This is an excellent question giving a really incisive inquiry into causal reasoning in a "simple" problem. The issue here is that when you are playing the Monte Hall game, you are making a predictive inference only --- you want to use the information you have to predict where the car is placed. You are not attempting to make a causal inference about the effect of you choosing a door --- i.e., you are not seeking to determine whether or not your choice of a door causes the car to move.

A causal inference in the Monte Hall game would be asking whether or not you choosing a door caused the location of the car to move (e.g., an assistant quickly and quietly drove it to another door based on hearing your choice). If you want to know this then you would not condition on the collider. You would compute the probability that the car is behind a particular door, conditional only on you having chosen that door and without Monte opening another door. Suppose you conclude that this conditional probability is $\tfrac{1}{3}$, which matches the prior probability that the car is behind that door. Since these probabilities match, you would infer that there is no causal effect --- i.e., your initial choice of a door had no effect on the location of the car.

A predictive inference in the Monte Hall game is the standard question we are asking --- given all the information we have, what is the conditional probability for the position of the car? Since we are making a predictive inference we condition on all available information including the collider variable, and this leads to the famous solution you have outlined.

As you can see from the above, the Monte Hall problem actually is consistent with the general advice given by Pearl --- when making a causal inference we do not condition on the collider variable. Another way of looking at this is that if we are making a predictive inference we don't care if a statistical relationship is "spurious" or not (see my related answer on "spurious" correlation).

This is an excellent question giving a really incisive inquiry into causal reasoning in a "simple" problem. The issue here is that when you are playing the Monty Hall game, you are making a predictive inference only --- you want to use the information you have to predict where the car is placed. You are not attempting to make a causal inference about the effect of you choosing a door --- i.e., you are not seeking to determine whether or not your choice of a door causes the car to move.

A causal inference in the Monty Hall game would be asking whether or not you choosing a door caused the location of the car to move (e.g., an assistant quickly and quietly drove it to another door based on hearing your choice). If you want to know this then you would not condition on the collider. You would compute the probability that the car is behind a particular door, conditional only on you having chosen that door and without Monty opening another door. Suppose you conclude that this conditional probability is $\tfrac{1}{3}$, which matches the prior probability that the car is behind that door. Since these probabilities match, you would infer that there is no causal effect --- i.e., your initial choice of a door had no effect on the location of the car.

A predictive inference in the Monty Hall game is the standard question we are asking --- given all the information we have, what is the conditional probability for the position of the car? Since we are making a predictive inference we condition on all available information including the collider variable, and this leads to the famous solution you have outlined.

As you can see from the above, the Monty Hall problem actually is consistent with the general advice given by Pearl --- when making a causal inference we do not condition on the collider variable. Another way of looking at this is that if we are making a predictive inference we don't care if a statistical relationship is "spurious" or not (see my related answer on "spurious" correlation).