As for what is theoretically possible with D&D? Well, 4e says it all! 4e, if you take the various parts where it says 'how to play' seriously, IS a Story Game. I mean, it is a bit broken, it isn't a pure story game. It is close though. It can be played, legally within the stated rules, process, agenda, etc. as such. And it works. I know, I did it for 10 years.
@pemerton did it, I'm pretty sure
@Manbearcat did it, etc. etc. etc. So, you have to go back and abstract out those elements, clarify them a bit, dig the 'hooks' in a bit better and more explicitly, and then get rid of a few elements (like stand-alone checks) which work against it. I've conveniently written that game (partially and somewhat inexpertly, but it is good enough that I can run it). Even my version really needs a rewrite because I didn't START OUT understanding all of this 5-6 years ago to the degree I do now. I have much refined the agenda of my game, and a lot of things don't fit well anymore, etc. Still, it is clear to me how to write a follow-on game to 4e that would be 'Story D&D'. Frankly I'm not even really concerned with things like pseudo-Vancian casting vs A/E/D/U and whatnot, those are all secondary. What matters is the core 'resolution loop' of the game.