The best way I've seen the monty hall problem explanation improved upon was by increasing the number of doors. The difference between a 1/2 and 1/3 chance is so subtle to a human. Change it to 1,000,000 doors and suddenly it becomes very obvious why your odds improve on swapping.

Tbh it was never very obvious to me why we reveal N-2 goats instead of just 1.

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.

This actually helped. I always assumed that the goats/prize were randomly placed, Monty's Door selection was random, and the contestant chose a random door. In which case I could never fathom why you'd need to switch. Since as the author says, there's no new information about the remaining two doors, and switching or not switching remain random.

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.

To be fair, someone who understands the full problem would be able to explain both solutions.