Using your analogy, if DnD is just rolling the dice then time wise it is effectively the commercial in-between the program of talking about stuff that separates you from the next DnD commercial.
Look, this is nuanced, and all-or-nothing analogies don't do anything but muddy the waters.
It has both, but really - which has more power to determine the outcome? The dice or the story?
I'd say in FATE, Apocalypse World, Polaris, etcetera - true story games - when dice are used, they are not to resolve in a granular fashion. They are used to force branching in the story - but it's mostly story. The rules for resolution are extremely simple, and the power to narrate how things happen, why, what they appear to be, etc. are almost fully under the players' control. In fact, it's explicitly collaborative, and that collaboration has no dice roles or rules for the vast majority of that resolution, so really, the dice are seen as a creative tool - a thing to do at certain points to force the players to improvise, but do so in a way that they see fit.
Those are fun - I really enjoy them - so I'm not judging like some do. I'm just calling them what they are - Story, with game elements to service the story. You can play out an arc anywhere from mostly to exactly the way you want.
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D&D isn't like that. It has tons of story that 'glue' things together, but when it comes to conflict resolution, the system is very defined, very crunchy, detailed, and that control is out of player's hands. They have
odds of success, and since the game is more 'heroic' than it used to be, they have more mechanics to overcome them (like the Luck feat, Halfling Luck, Inspiration, etc.) but those resources are still limited and eventually that luck runs out.
So, when an attack happens, you have much less control - you were on the way to slay the Dragon, but an avalanche in the mountains caught you on the way there - so roll your save, and if you fail you die - and you can't change that. Instead of your story being one of 'Father was slain by the dragon and got revenge by delivering the killing blow', it's now 'Father was slain by the dragon and died before encountering it'.
You didn't choose the way it ended. Ultimately, the encounters and the dice did. You told the story as how
this person with
this origin dealt with these situations - that were resolved by dice and fairly complex rules - after the fact.
This is true for combat, and now (in 2E and up) in the social realm as well. No matter how eloquently you role play, a Charisma of 6 will have an absolute effect on how effective your character is in interpersonal relationships. No more 'drop stats' that are nullified by skilled players; the Ability Scores hugely influence your fate.
That's not to say there is no story, because how your characters deal with each situation is going to have a huge influence on how they work out, and whether your character succeeds in their goals or not, but the thing is, dice rolls have the ultimate authority in conflict resolution unless you house-rule them away.
DMs can jump in and nullify rolls, both good and bad - it's your table, you have that right - but if you do that, at a certain point, it's no longer D&D. The game has death. It has failure. It has social failures that can shape relationships. It has cursed items, ignominious deaths, and the possibility of total party wipes.
If you arbitrarily decide 'no, we don't die' then you arguably have made it in to a story with game trappings, rather than a game with story trappings. And that's fine, but that's not the game as written any more, and your judgment as to what D&D is or is not is moot - you're judging your own house-ruled game, not D&D. Even in it's most recent 'heroic' incarnation.
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TL,DR:
In the balance of Game vs. Story, D&D 5E falls in between, with a lot more emphasis on character and story, but when push comes to shove the game as written still relies on dice. Subjectively, that can vary a lot from table to table, but since dice still ultimately resolve conflicts, it's a game - one that, afterwards,
results in a story.