Sure, I see what you're saying. In a technical sense the information in the host's head does "matter". My comment is more along the lines that "information content in a system" is a sufficiently abstracted idea from "knowledge of a human being" that, especially if you equivocate them, it's not a useful metaphor for someone who doesn't already get why the right answer is right.
Yeah, it's not even strictly "does Monty know?" It's whether his action embodied that knowledge or not. Obviously (I hope) Monty knowing but still picking based on a coin flip is the same as Monty not knowing.