In a game, I’d say there’s a really wide spectrum on this issue, and readers should try to align their expectations accordingly (though of course taste is different than expectation, and one’s taste may be decidedly for one end of the spectrum or the other – as both yours and DavidGil’s seem to be).
Some games are written so the playable character = you, the reader. Or whoever you, the reader, wants the character to be. DavidGil has described this as “generic” in a couple of places, and I think that’s often true – but I also think that’s because the reader is being implicitly drafted in as co-author, filling in the details, and that the character as ultimately experienced will not be generic because it will be personalised in imagination. Just read some of the threads here for e.g. Choice of Robots or Guenevere to see how non-generic people’s conception of their characters ends up being.
In other games, the playable character is definitely not you. It’s Leisure Suit Larry, or Tommy Vercetti, or the dad with the dying kid. If you come to this type of game with the expectation of control, you’ll be disappointed – as DavidGil notes that a number of players of Telltale games have been.
In this type of game, you’re implicitly being asked to act rather than write. Act out a character who is not you, or chosen by you… who will not always react as you would react under those circumstances. If we can get our expectations in line, those games can offer potentially a more powerful (and if the game is good enough, profoundly unsettling or revelatory) experience, taking us out of ourselves and letting us see a world through the eyes of the Other.
That said, I’d also note that a game with a generic playable character doesn’t have to be a game with a generic main character. Bioshock is the story of Andrew Ryan, not the weapon-wielding fists in the foreground of the screen. Myst, jumping back a few years, was the drama of the Book Brothers rather than their mute observer. And both of those games are of course world-driven as much as character-driven, where the settings have a rich and unfolding story. I’d argue that there can be great literary as well as entertainment value in generic-PC games, too.