This is interesting but it strikes me as culturally fairly far removed from the common culture of D&D play. I wonder are there games have have elements that encourage narrative that are closer to the cultural norms of D&D that could be added as an optional extra for those tables that are interested and willing to buy in.Probably the single best-known "narrative" game is Apocalypse World. It's the game that inspired PbtA.
AW does not have particularly fancy mechanics. It doesn't have fate points. It doesn't use "say 'yes' or roll the dice" (that's from a different Vincent Baker game, DitV). It has a very small number of PC build options which allows players to establish fiction which goes beyond the in-fiction causal consequences of things their PCs do.
There are two main features of AW that mean that, in play, it yields story.
The most important is the GM moves. When it's the GM's turn to speak in the conversation of the game, the GM make a move. And most of the time this is a soft move. GM soft moves include (inter alia) "announcing future badness" - in ENworld parlance this is a form of what is often called "telegraphing" - or "announcing offscreen badness" or "offering an opportunity, with or without a cost". In some circumstances the GM can make a hard move - this can include (inter alia) "inflicting harm" (ie dealing damage) or "separating them" or "turning their move back on them" (eg the PC as played by the player tried to discern a weakness, but instead reveals a weakness of their own). The list of GM moves does not include nothing happens.
Nearly as important are the player moves. These are little resolution subsystems that are triggered, in the course of play, by the player declaring a certain sort of action for their PC (eg trying to intimidate someone ("go aggro") or trying to exert leverage over someone ("seduce/manipulate")). If a move fails, the GM is licensed to make a hard move. The other time a hard move is OK is when a player hands an opportunity to the GM on a platter (eg the GM makes a soft move, the player proceeds in disregard of it, now the GM can make a hard move).
If a player declares an action for their PC that does not trigger a move, then no dice are rolled, the conversation of the game continues, and the GM makes an appropriate move - probably a soft move, unless the player's declared action provides an opportunity on a platter.
This is fairly different from a lot of D&D play. Not because of anything about the mechanics. But because of the rules about what the GM is to say.
It seems to me that "pushing the game toward story" as the thread title suggest involves a change in the culture of the game. An barring a CR like popularity for Apocalypse World this can only be accomplished by iterative nudging of the base D&D experience by the addition of tools or other support (Let's play videos) or something that opens up the experience.