The analogy is not chosen merely for being similar, but to fulfill a purpose. It's based on the idea that in the original puzzle what you are faced with is a choice between your original door, or all the other doors. That's why you have a 1/3 or 2/3 chance of being right if you stick or switch respectively - because you are either choosing one door, or effectively two doors. The host showing you whats behind a door is just an implementation detail that causes you to have a stick or switch decision that has that effect. That's why a nice analogy is one that gives you a choice between 1 door or 99 doors, i.e. a chance of 1/100 or 99/100 of winning.
Your variant is a valid analogy too, but the probabilities in your scenario between sticking and switching are so close to each other, it's not going to trigger any new intuition in someone who is already confused.
Don't think of the 100 door puzzle as an analogy with the 3 door puzzle, think of it as a class of puzzle where there are N doors, you can pick one, and then the host (who knows what's behind the doors) reveals N-2 fails. It's clear that the 3 door puzzle is a specific instance of that, and if you have understood something intuitively about that puzzle class through considering N being a large number, that will still hold for N being a small number.
The class of puzzles that you describe is M doors, where you pick one and then the host (who knows what's behind the doors) reveals one fail. The three door puzzle is also a specific instance of that puzzle, but I doubt that you've learned anything from that, since examining M at other numbers probably hasn't given you any insight (still, if it did, those insights are indeed applicable to 3 door Monty Hall).
I've tried a number of ways of explaining this problem to people, and empirically the 100 doors one is the way that seems to work best. Interested to hear any versions you tell that seem to help people understand whats going on.
I've been wondering about another somewhat intuitive way to see the problem: suppose a third person enters the room and he is allowed to finish the game, knowing nothing of what went on before. In particular, he doesn't know your initial choice. Now it is obvious that his chances of winning are 1/2. It is also obvious that if you stick to your first choice, your chance of winning is still 1/3. Now it is odd that somebody who has LESS information about the problem has a better chance of winning than you. Meaning that you must have some way of getting at least the odds of the other guy. But there's only one thing you can do other than sticking to your first choice: taking the other door. So that seems the correct thing to do.
Your variant is a valid analogy too, but the probabilities in your scenario between sticking and switching are so close to each other, it's not going to trigger any new intuition in someone who is already confused.
Don't think of the 100 door puzzle as an analogy with the 3 door puzzle, think of it as a class of puzzle where there are N doors, you can pick one, and then the host (who knows what's behind the doors) reveals N-2 fails. It's clear that the 3 door puzzle is a specific instance of that, and if you have understood something intuitively about that puzzle class through considering N being a large number, that will still hold for N being a small number.
The class of puzzles that you describe is M doors, where you pick one and then the host (who knows what's behind the doors) reveals one fail. The three door puzzle is also a specific instance of that puzzle, but I doubt that you've learned anything from that, since examining M at other numbers probably hasn't given you any insight (still, if it did, those insights are indeed applicable to 3 door Monty Hall).
I've tried a number of ways of explaining this problem to people, and empirically the 100 doors one is the way that seems to work best. Interested to hear any versions you tell that seem to help people understand whats going on.