Basically, because they can't.
To try to use the video game medium to tell a story you need to be extremely good at what you're doing. Since you're playing the main character... why does one always have to play the main character?... you are always in a position to control the action. However, you almost never get to control or affect the story of a video game.
Characters very seldom react to what you're actually doing while controlling the character. You cannot complete different objectives than the ones you are given. You cannot save the person you know is going to get their ass kicked in a couple of minutes. You are, as a player, completely powerless over any form of story.
Since you're given a story and enabled to interact with the world but not with the story, you start to feel detatched from it. It cannot swipe you off your feet because there are too many moments where the illusion is broken, where you are no longer the character, but the guy sitting with a controller in hand.
Imagine a movie. In the middle of the most intense actionsequence, the hero chasing the villain falls and twists his ankle. The villain gets away. The director shouts cut, walks into the picture, tells the main actor what to do, and they do it all over again. This would likely leave you with a sense of detatchment, that this is just a movie and not something important. No?
Or imagine that the Hero knows that the Villain is going to shoot his best friend in a couple of seconds, but he doesn't do a thing about it.
This, I claim, is what happens in games ALL THE TIME!
As soon as players are allowed to mess around with things and the game does not respond to ALL of these things in a believable way, the story will get less and less important.
It's not just confined to Anime/Manga games. It occurs in ALL games (wow, that's three ALL's in caps (well, four now)). It is however extremely apparent in games that build on something that is known for its strong story.
There are a couple of ways to limit these effects.
1. Limit the players freedom in world - Case in point, Pheonix wright games. Amazing story, absolutely no freedom or control.
2. Build a world that responds to as much as possible - Baldurs Gate, perhaps. Mass Effect.
3. Make the story such that any response is unneccesary - Portal.
4. Build a world that acts only as a representation of something bigger - Civilisation and Total war games do this. Since the interactions are all on a macro level our imagination fills in the blanks of what must occur on the micro level.
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