My experience (not too surprisingly, I suppose
) has been almost completely opposite. I get tired of unveiling "the story of this guy/gal" and listening to him talk. I don't find it interesting for its inflexibility and because it's usually very predictable. And once I've seen it through once (if it is even good enough to bother wholly through), it can be tossed away as there's usually no reason to see it again.
I don't get as much 'personal' high drama with an open PC, but I don't need that from a game (to be completely honest, I don't really want it). It's a game, so the first thing I want from it is as much player agency as possible in both gameplay and narrative. I don't think the story needs to be anything but an initial motivation (a strong one though) to set things in motion for the character and a reactive set of conclusions as per how you followed that motivation; between which you weave your own path (that the game recognizes, not like in Bethesda's games where nothing matter and nobody cares, just go do stuff) as much as the game allows. There of course needs to be strong writing, but strictly following that writing shouldn't be the central point, letting the player do his thing with it should.
I find it much more intriguing to try out what all I can do and what all ways I can tackle the narrative, that I have a say in these things, rather than following a specific someone's specific storyline.
This is what makes Fallout (the originals) and Arcanum such brilliant games. There
is a strong sense of narrative, but it is all up to you how to go about it... and the game doesn't just set you loose witha character of your choice, it keeps track of what you do and how and responds.