Given that I know from experience that at least two versions of D&D can be played "story now" - AD&D and 4e D&D - I don't think a contrast between D&D and "story now" sheds much light.I'm not sure there's much if any difference here between story now and, say, D&D. In fact, one of the sometimes-criticisms against D&D is that the stakes are often too small and-or the system is too granular; yet here you seem to be advocating for just this. And I agree with you.
And in both systems the small things can feed off the large; for example, who's going to look after my farm animals while I'm out on crusade? The crusade represents the larger stakes, and all the smaller-stake items arise en route to getting at the larger one...if that makes any sense
Most versions of D&D don't do intimate stakes very well, though, at least in my experience, for this reason that I've mentioned a few times:
It's helpful, here, to know how your game's action resolution system works, because if you prompt your players to declare actions that your system can't handle, that can be a problem. It pushes play away from the player protagonism you're aiming for, and into either rules debates, or rules-free storytime.
In the example of looking after the farm, How will it be determined who is given this job?,and How will it be determined whether or not they do it properly and How will it being done poorly be brought home to the character?The main constraint, I think, is not the scope of the fiction but rather what can your chosen system's action resolution rules handle.
The versions of D&D I'm familiar with struggle to provide answers to these questions. I'm happy to be contradicted by others' experience, but I don't think it would be easy to build an engaging skill challenge around these questions. And a fundamental problem in D&D is that, given the way the game handles character development and progression of capabilities, it is likely that whatever has happened to the farm when the PC returns can be set to rights by them with little difficulty. So it turns out not to have really been at stake at all.