Personally I don't believe stories in video games should be presented wholly for the purpose of "moving the action along" & any game that does this has missed the point entirely...
I think first and formost games' stories so far have existed only to provide the player with a sense of purpose.. To provide some form of constrained play experience which has a beginning, middle and end.. To provide the player with a progressive challenge which allows the player to move from one challege to the next and ultimately reach an end point where the satisfaction & rewards are offered..
This mechanism however is not something in reality which "requires" a "plot" to perform however & many games (predominantly the titles of old) have provided the player with that same progressive play experience (challenge->reward->challenge->reward) without the need to implement a strong narrative mechism..
In my view Half Life 2 however is a great example of a title which truely uses a plot/narrative/story mechanism, not as a driver for progressive play, but as the fundamental basis which defines the purpose of the game in itself..
I truely believe that stories in games can reach new heights with regards to capturing the true essence of the narrative for the user & really exploding their imagination in terms of presenting a truely compelling & believable world however I believe that the mechanisms used to do this need to evolve.. Sure the technology is there now & it can really aid in provideding new tools to help (human expression/digital actors like in Heavenly sword being one..) but how these tools are used ultimately defines the quality and ability-to-captivate of the whole experience..
It's not always the most visually descriptive elements that contribute to telling a compelling story & even games like Warcraft (a 2D RTS being probably the most abstract gameplay mechanism one could use to really engross the player in a world/plot/experience) used very subtle elements to really engross the player in a believable world full of enchantment, history and lore you could actually care about..
I think that purpose is the only reason a story is used in games right now. It could be used in other ways I suppose, but it's quite difficult to blend it with engaging gameplay at the same time.
HL has atmosphere and great pacing, not a great story. In the end the plot could be anything, the presentation is what drives the game forward. The same story with non interactive cutsceens would be IMO very uninteresting.
Besides, purpose in a game is not a bad thing, if the player wants to acomplish it. It is however very difficult to force the player to care about the given purpose. In Heavenly Sword for example, most of my friends skipped the cutsceens whenever they could. They wanted to kick some ass, not to watch an animated drama. On HL though, it's in your face, you have to pay attention.
As you said, games require a beginning a middle and an end. A predetermined path that unlike film or book you take active part in. It is very difficult to make someone care when he knows he can reach the end whether he is paying attention to the plot or not. Perhaps if his actions had severe consequences he would be more interested.