Regarding video games, I think it's too diverse a medium to really discuss what it's "supposed to be". It's great a storytelling, but that doesn't make it a "storytelling medium" any more than film being good at making people want to buy stereos makes film an "advertising medium".
I'm probably splitting hairs here, but here goes.
You'll be hard pressed to find a game that doesn't either A) have the player act out a story written by the designer, or B) give the player opportunity to act out their own story.
Examples of A are obvious, but examples of B are (almost) every other game in existence. Dwarf fortress for example has no set narrative, but as that one new york times article says,
Tarn envisions Dwarf Fortress, by contrast, as an open-ended “story generator.” He and Zach grew up playing computer games with notebooks in hand, drawing their own renditions of the randomly generated creatures they encountered and logging their journeys in detail. Dwarf Fortress, which never unfolds the same way twice, takes that spirit of supple, fully engaged play to the extreme.
Even games like farmville, where the storytelling elements only exist to give context, has
some sort of narrative, either player or designer defined. You're a farmer, wanting to build a bigger farm. Not the greatest story, of course, but still follows the definition of a story: there's conflict and a drive for resolution. Even the most mundane games need their excuse plots.
Now, there do exist games with absolutely 0 non-meta storytelling elements whatsoever. The example I usually give for this is tetris. There's no reason for the blocks to be falling, nor a reason for them to disappear. As far as we know the blocks are just abstract representations of... something. Maybe the rise and fall of soviet union.
So you'd be correct in saying storytelling is not completely
intrinsic to the medium, but it permeates the vast, vast majority of games. Only things on the very top of
this exist without story (and honestly I'd put tetris above pong, as pong has at least context).
You'll also be correct to say that story "isn't the point" of many, many games. They may have an excuse plot, but it just exists to support what really matters. To that I'll say there exist films (and other things) where story "isn't the point," either. Look at cheesy action flicks, where the viewers are simply there for an adrenaline rush, not because they care about character interaction or anything. Does that mean films aren't a "storytelling medium," because there exist films where storytelling isn't the focus? I don't think so. Nor do I think video games not a "storytelling medium" because there exist games where story isn't the focus.