I disagree with this idea completely. I can't stand playing games where the main character is afraid of every little thing, like Amnesia. Cowering and shaking don't make a game immersive, they make the player feel like the main character needs to grow a pair.
Hahaha, I hated that about amnesia. A slight breeze and your character was quivering with fear for a minute and the whole time you just wanted to keep moving on with the game.
I don't think that being afraid of everything is a good way to provide a more deep experience for the player at all. It's a decent suggestion but is better suited to games that center on the aspect of being afraid, and even then it has to be done properly.