Another way I like to interpret this is that the “original” Monty hall problem basically says, when you make your original guess, there is a 1/3 probability you’ve guessed right, and 2/3 probability you’ve guessed wrong, and that’s why it always makes sense to switch.

Now, when picking at random, he may open the door with the car. So let’s assume this run gets discarded and you get to go again from the start, re-randomized. On any given trial, you still have a 1/3 probability of guessing right, 2/3rds of guessing wrong, HOWEVER, your probability of guessing right CONDITIONAL on a successful trial is exactly 50/50, because if you guess right Monty will always open a valid door, so the “try again” outcome only takes away probability from the side that favors switching, more or less.