You forgot that all of the doors have 1/3 before the host opens a door, so both the contestant's door and the mystery door are 1/3 before the host shows the goat which makes it seem an arbitrary choice to assign the mystery door 2/3 after the reveal. Why not assign it to the contestant door? The fact is it shouldn't get assigned to any door. The game was never between three doors.
No, that's not what I'm saying, and I think you're making the mistake I highlighted in the 1000000 doors reductio ad absurdum. I'll try to make it clearer:
Assume for the moment that you're a computer, and you don't care what anyone says, either before or after you choose your initial door (at random).
Each door has a 1/3rd chance of hiding the car a priori. This means that any initial guess, has that same chance of 1/3rd of getting the car - it doesn't matter which door you pick.
So now you've made your guess. Your door has a 1 in 3 chance of being right, and there is a 2 in 3 chance that the car is behind one of the other doors. Do we agree so far?
OK, now one of the remaining two doors, which taken together have a 2/3rd chance of being correct a priori, is opened, to reveal a goat.
Being the computer that you are, you infer that this means that, whereas the two doors together have a 2/3 chance of containing a car, the one that was just opened has a 0 out of 3 chance.
Still. the remaining two doors MUST have a shared probability of 2/3rds, because your initial
guess cannot have had a chance of more than 1/3rd of containing a car (assuming the car is behind a random door). This is because you don't change your initial
guess after the fact; you made your choice while there were still three doors
, so it had 1/3rd probability, and this cannot change unless you have a time machine.
That means that the remaining 3rd must shift to the remaining door, giving that door a 2/3rds probability of hiding the car. You still have that 1/3rd chance of being right that you had all along, but the odds are against you.
Try doing the 1000000 door substitution on this narration, or even a 4 door substitution (remember that Monte opens all but one door, rather than just one extra door. That cognitive flip helped me a lot when I was struggling to understand it).