> opens another door, say #3, which has a goat

There is no randomness in this sentence. The door has a goat, not a car.

Then again, you might read it as an example outcome. But even if the door was chosen randomly, it would not justify the 50%/50% answer.

Independently of that, assuming that the host chooses the door with the car and the goat with 50% probability each is the same mistake that confused so many, you cannot always assume uniform probability only because there are 2 options.

(I wanted to point out the flaws in your thinking because this has also been a confusing problem for me)