It depends entirely how the story is told, and the how the dialogue is written. In a game like Half-Life it works because Gordon is just an observer to the events that are an unfolding; he's not expected to take on a specific role or respond in a certain way, and the character itself is meant to be an enigma (how does an ordinary scientist manage to kill all those people?). Essentially, Gordon is a vessel for the player to inhabit for the duration of the game.
Compare this to something like Metro 2033 where the player's character speaks between each chapter, and is involved in 'conversation' (where the game implies that the player character has spoken previously, but doesn't explicitly show this), but completely ignores the other characters, during gameplay, even when they make direct requests to him.