I don't see how a new contestant has less information, though? They know that one of the two doors contains the prize, which is all the previous contestant knows either.
If the game had different rules, it would work like you are imagining. Specifically, if Monty randomly eliminated one of the two doors, meaning there was a chance for Monty to reveal the prize instead of a goat. If Monty has the chance to eliminate the prize before giving the contestant a chance to switch, then switching does not give you an advantage.