Confusion about the Monte Hall problem isn't an issue with people not understanding probability, it's that the problem is usually presented in a way that's under-explained and with hidden assumptions:
My own explanation is that if you are shown a goat door in advance, there is 50/50 of two remaining doors. But since without that you might have picked the door that was revealed, there’s a symmetry about it in which the swap is more lucky. Which is just a clumsy way of saying your chances were 2/3 lose vs 1/3 win and it doesn’t change after the reveal, so you must swap.
I still feel like I don't full grasp the actual problem/solution though. Even with this new understanding. I will read the linked article by Jeffrey S. Rosenthal though, and hopefully that will fill in the last of the blanks.