The other phrasing of the problem that I've seen help people is this (I think was also originally from Marilyn Vos Savant): Suppose there are a thousand doors, and only one is a car. Monty opens 998 of them to reveal goats, leaving door #628 still hidden. Do you switch?
That helps people understand that their original probability of picking the car the first time was very small, and didn't change because of what Monty did.
Most paradoxes vanish, become obvious this way (when you increase the numbers). The brothers paradox mentioned: 'I have two children. At least one is a boy.'/'I have two children. At least one is a boy born on Tuesday.'; 'What is the probability I have two boys?' drastically changes when you change to 'I have two children. At least one is a boy born at 7h:35m:42s.' -- it becomes clear that the succeeding information exactly specifies the boy as it becomes specific, in the limit turning into 'I have two children. The first is a boy.' (so the probability goes from 1/3 to 1/2).
You pick a door. Monty opens another door, revealing a goat. You now get the option to switch from your door to the other two doors, keeping the car if it's behind either one.
Whether Monty opens the door or not doesn't mean much.
That is only equivalent if Monty (as intended but usually not stated clearly) always picks a door without a car using knowledge of where the car is. If, instead, Monty picks randomly from the other two doors and just happens not to have revealed the car this time, odds are the same whether you stick with your door or switch.
Even in the case where your restatement is mathematically equivalent, it's obviously not at all equivalent as a puzzle because what is being requested by the puzzle is seeing that the original situation is mathematically equivalent to that one.
That helps people understand that their original probability of picking the car the first time was very small, and didn't change because of what Monty did.