I think you'd need to expand this train of thought a little if you want it carry any water. I suspect that you'll run into trouble when it comes time to try the experiment with 'traditional'. RPG mechanics do all kinds of things, not all of which directly influence the conversation as such - there is also the setting and the player avatars.
While you can certainly identify some mechanics with a narrativist playstyle, I'm not sure you could find one that defines the style. Or even a few that do so. Perhaps a list of types of mechanics might do that job though.
I'm not quite sure what you mean by 'doing that' there. In addition, this happens in a lot of different parts of the game, and in many kinds of games, and to many degrees. I don't think you can point to category of games that do it. For example, The Black Hack uses player background pretty significant to build out the setting, but it not a narrative game. In fact, lots of OSR-adjacent games have added mechanics of this sort but without changing the overall telos of the game. It simply isn't a useful binary.