Can you give an example of a case where being predictable would hurt someone who goes about choosing actions well in the first place?

Good question. Your intuition is correct as long as your actions are chosen "optimally" in the game-theoretic sense. This is one of the ideas behind Nash equilibria: your opponent can't gain anything from knowing your strategy and vice versa. A caveat is that the Nash equilibria of many games require "mixed strategies" with unpredictable randomizing, so if the opponent can predict the output of your random device, you're in trouble.