If Monty knows the door with the prize and is aiming for the game to continue, then you should switch. (This is the usual argument.)

If Monty doesn't know where the prize is, then you learned nothing. (Monty's result was luck, and he can't impart information that he doesn't have.)

If Monty knows where the prize is and wants you to lose, absolutely don't switch. (Monty will only drag the game out as a way to try to make you lose.)

The reasoning behind these statements is completely solid, and there are no hidden assumptions being snuck in.

You can't just sneak in the assumptions. You have to state them somewhere.

You specifically need information that we haven't been given about the counterfactuals. What might Monty have done in other scenarios that we have not yet observed? We don't know. And we're not actually told. That makes that an implicit assumption that wasn't specified.

Edit: Furthermore, I don't think that author's solution to the Crawl problem is correct. When the host eliminates a door, either you will get information that says you should switch, and you'll win 100% of the time; or you won't get information, and you should still always switch and win 2/3 of the time.

That's the missing assumption. I would say assuming that people are perfectly random falls into the "standardized test" category.

> you won't get information, and you should still always switch and win 2/3 of the time.

You always get some information, the set of possible results becomes narrower, so saying the probabilities don't change is not sufficient. Not a good idea to discuss the problem in informal language though.

That was when I realized many people just memorize the Monty Hall's solution without understanding it, a.k.a. "standardized tests".