When the Monty Hall problem was first brought to my attention a few years ago, I didn't get it right either.
But here's the thing: perhaps because I hadn't seen the show for decades, or else because I was taking the reference to Monty too literally, I was thinking about him as a strategic player who wants to trick you, because it's fun for the TV audience to laugh at you as they play that "wah-wah-wah-wah" music once you've switched from the new car to the smelly goat (or whatever).
So I figured, he is more likely to show you a door that is the wrong answer, and then to ask you if you want to switch to the third door, if the door you initially picked was the right one. Hence I hadn't accepted the (often unstated) premise of the Monty Hall problem, which is that, no matter what, he will show you a door that is the wrong one.
Once you introduce the possibility of situation-specific strategic play by Monty, rather than his doing the same thing every time, there is no right answer to the puzzle until we know more about his decision function.
This causes me to think of the Monty Hall problem as demonstrating, not just our difficulties in thinking clearly about probability theory, but also our inclination for strategic thinking (and imputing it to others). That surely is highly adaptive as a general matter, whereas tending to bungle probability theory is presumably more about heuristics and cognitive shortcuts.