I like to think of mechanics as the layer of affordance. The level of interaction we can get out of a game depends entirely on the player input and how that is fed to the mechanics (either directly or through UI (there’s a big difference between these two but let’s not get distracted (oohh a butterfly!))). All first person shooters are about killing, because that’s what you’ll be doing the majority of the time: shooting dudes in the face. That the main protagonist has a kidnapped daughter who likes the smell of pizza in the morning doesn’t matter … because that’s how we build these games.
If someone were to describe a game to me explaining only it’s context and how meaningful it’s themes are, I’ll immediately ask about what type of game it is. Is it a shooter? a puzzle game? an IF? We are so used to having games with stories divorced from game mechanics that we don’t find it odd when a game about objectivism in a post apocalyptic future turns out to be a game about shooting zombies in the face with superpowers and guns.
Story in itself is most of the time used to provide a context to the mechanics and may very well make a big difference in how a player feels about performing or not performing a certain action. But it has to go both ways for it to be effective: The story has to inform the mechanics and the mechanics have to revolve around the story. You can’t have a game about shooting people in the face for 20 hours and expect to touch meaningful themes that don’t have anything to do with the shooting itself. Yes, you shot that guy’s wife and now he’s a little bit upset, but that doesn’t mean that the game is suddenly about loss and grief.
“Show, don’t tell” applies to movies just as much as “Do, don’t show” applies to games. For a game to be about something it has to happen to the player and the player has to be the one acting upon said something. Revenge stories work in games when and only when the player has had a chance to interact with the soon to be excuse for violence. Play a few hours around the house with your family before they are inevitably kidnapped or shot at point-blank range (… in the face).