Let me make an analogy with Rubik's_Cube. What makes that fascinating, and hard to solve as a puzzle, it that it's different from familiar physical objects. It's an artifact, designed to be different and challenging. Analogously, a key feature of the Monty Hall problem is that it's quite different from other "decision under uncertainty" problems; one cannot imagine any closely similar problem arising, outside the context of a game with rules invented to make it challenging.
It is true that the Monte Hall problem is one of a standard list of mathematically elementary toy problems that one easily gets right by using the formal setup; but one can create a less standard list of slightly more real-world problems for which the formal setup leads you astray: here is one example. The basic point (discussed at greater length here: xxx to be written) is that in the real world we get information that has been selected in ways we can't analyze. In Monty Hall there are only three equally likely initial possibilities and an agent acting according to a known deterministic rule, so we can do the analysis. Can you think of any real-world story, outside of an invented game, with this simple a structure?