I am in no way ready to defend 5e's missing skill system, and I'm more upset than the average gamer that resolution has trended away from complete skill systems. I lay much of that blame on the rise of generic difficulty and difficulty by level tables, but that's a whole other discussion.
There's a whole under supported mode of play here for sure, likely because it has heavier design requirements.
I'm fine with the 5e skill system (modulus some greater concerns with 5e design in general). 5e would go a fair ways in terms of being suitable for narr play if it recast checks in general in a way more like PbtA where you check something, and there's a hard and fast rule that THINGS MOVE ON, either the player gets what they were attempting to do (at a fairly atomic level, granted) or there was a consequence of some sort. A 'mixed success' result might be nice too, but maybe it isn't needed, I'd have to think about it more. However, you still have the transparency problem. So, I want to steal the papers from the Duke. As 5e is written I don't really know what that will entail, its all hidden GM-side information. It could require 3 skill checks, or 30. The GM could rule any degree of consequences for any specific failure, total mission fail down to slight inconvenience. Nor is there even a mechanism or a suggestion to clarify that before the check is made. As a result PCs are flying blind into GM curated darkness. Heck, by taking away the DC-by-level aspect of 4e we don't even get a notion of how hard the checks will be, from simple to impossible, until we arrive at each situation!
Now, obviously, a GM could arrange a whole sequence of play that illuminates the "get the papers score" and gives the PCs some idea of what these parameters are, at least in fiction terms. Unfortunately 5e isn't super good about making it clear which fictions are really appropriate. Outside of combat there isn't anything even resembling a 'CR'. How do I even do something like balance the options of fighting vs sneaking in a given situation? It is all pea soup!
Narrativist play NEEDS these sorts of things to be present! I need the papers to save my sister from execution. So, what makes this interesting? The fact that I can, as a player, look at the situation and say "I am pitting myself, as the character, against almost certain death to save my sister!" Well, that's pretty substantive. Granted, in the real world everything is muddy and grey, and you just don't know, or can only suspect. But that's not what we're after in RPG play! If its a grand gesture, it is spoilt if nobody knows that!
I mean, this is all just examples, touching the surface, but you see there's a pretty steep road to climb to get something approaching AW out of 5e.