I hope I'm not barking up the wrong tree here. Should I have waited for you to start your own thread?
Around here we are big believers in the idea that if you have a theme or prose, the mechanics needed to support it becomes apparent. For this game, you want quick action, but you also want a story. All of these sorts of games have that, so what do you specifically mean?
Are you looking for a way for players to influence the environment or scenes? Or perhaps you are trying to find a way to enforce a story arc?
For something like that you might want to consider another game on 1km1t.net called that uses a scenery deck - players get a chance to play cards that add elements to scenes. That game also grants each player a theme tune, and they get bonuses based on how they incorporate it. (A science while it plays, quoting it in conversation, and so forth.)
Another way to do this is to use a poker deck, with various suits representing allies, locations, equipment, or other benefits. Each person gets dealt a hand, and may play them to alter a scene in progress. This change can be countered playing another higher card, so you have some strategic management of your resources. In turn, the players explain why all this matters (yes the villain has a cache of dynamite, but the fuses are too wet to burn because of the environment card...)
For the story arc version, provide rewards or benefits for creating a plot. One idea would be to give players limited points to spend that are only replenished when they add a new plot-twist or danger to the story. A pool of hero points can grant extra dice or buy allies, but if they don't leave town, they're going to step on too many toes and run out of favors - so advance the story, or gain nothing back after use.
The Game "Warrior, Rouge, Mage" defines characters by how many points are put into those areas of expertise. A similar thing could work with concepts in your game. Perhaps the characters are rated by city/frontier/wilderness to get that trifecta. If not outright copying the WRM use of these as attributes, then what about making C/F/W the expendable resources mentioned above? Or perhaps, they are like hit-points for different types of challenges - a person with a high city rating does well with others and is far less likely to be kicked out of town for insulting the wrong people. A wilds character might hate dealing with people, but can shrug off dehydration and falling off a cliff because they're built to survive that.
What mechanics you actually use influence the story in many ways. If combat is not very dangerous to the players, stories will tend to end up rather more combat oriented. However, if there is a system that penalizes the character for ever occasion they use violence - they lose a part of their soul, or a family member, or a piece of equipment -then non-violence becomes the much preferred option.
Simple mechanics, tough main characters, and weak or incompetent minor henchmen inherently does lend itself to pulp style adventures - the question is what else do you do for the themes.