I believe the confusion with "ignoring the car picked cases" comes due to thinking that they are additional games to the original ones, instead of being part of those that constitute the 2/3 in which the player starts picking wrong.
So you thought that eliminating them you were left with the original 1/3 vs 2/3, when you actually removed half of the 2/3.
That's not correct, because if you agree that one kind of result tends to happen more than the other, then it is easier that the single game that you are currently playing belongs to the larger group. When you play once, it is like if you were randomly choosing a game of all the possible ones that you could play, in which case you are more likely to choose one of those that are more numerous.
It's the same reasoning that you would apply in other aspects of life. Imagine you have to travel to some town and there are only two possible roads: A and B. The stats tell that on road A there tend to occur 100 fatal accidents per year, while on road B only one accident has been registered in the last 10 years.
The causes of that disparity may be multiple, like, for example, that road A has many curves with precipices while B does not. But you don't even need to know those causes; the starts themselves tell you that as result it is easier to get a falal accident on road A than on road B. But, of course, it always exists the possibility that you survive in any of them, and also that you have an accident in any of them.
So, if you were to travel through one of them just once and had to choose, I don't think you would say that those stats don't matter only because you will travel once. You would prefer to take the safer road B.
In Monty Hall game, deciding to stay would be like taking road A, and deciding to switch would be like taking road B.
Moreover, if the game consisted in that instead of revealing a goat, the host gave you the opportunity to reject your first choice and instead check inside the other two doors and take which you prefer from them, it is obvious that it would be better to switch to the other two (unless you think 2 is not greater than 1), and that's true regardless of if you are playing just once or multiple times. Anyway, by checking those two doors you would necessarily find at least an incorrect one, as there is only one prize in total.
If you notice, as the host knows the locations of the contents and always removes a losing door from those that you did not pick, then it is like if he was doing that work for you of checking inside those other two doors, eliminating from them the incorrect one that you would have found anyway, and leaving closed exactly which you would have picked if you were who had checked inside those two doors.
So, if you agree that it would be better to switch to the other two doors, you must also agree that it is better to switch to the other single one that the host is offering, as both ways win in exactly the same cases.
So you thought that eliminating them you were left with the original 1/3 vs 2/3, when you actually removed half of the 2/3.