The fundamental role of the GM in PBTA is to decide when they need to roll, and secondarily, to play NPCs.
As someone who is preparing to GM Apocalypse World, that's not quite how I see it. The rest of this post isn't to quibble pointlessly with you - I've gor a reasonable sense from your posting history of the range of RPGs you've played/GMed, and am not meaning to interrogiate or cast doubt on your expertise.; I'm headed somewhere else which is a little bit OT, but seems relavent to the issue raised by
@DMMike.
In Apocalypse Word, determining when to roll - under the principles that
if you do it, you do it and
to do it, do it - seems primarily a table function. Perhaps the GM has a "chair of the committee" sort of responsibility - "there are two ways they sometimes don’t line up, and it’s your job as MC to deal with them" (p 12) - but that seems more like an outcome of the GM's role in managing the fiction.
It's that role of managing the fiction - helping keep everyone on the same page as to what is happening in the ficiton, and introducing new fiction when appropriate - that I would see as fundamental for the AW GM. (And as I read the rules and prepare, in my mind, for running this game it's the number one thing I'm thinking about!)
The prescribed agenda - "Make Apocalypse World seem real; Make the players’ characters’ lives not boring; Play to find out what happens" - is all about the content of the ficiton and the process of establishing it. The instruction to "Always say what the principles demand, what the rules demand, what your prep demands and what honesty demands" is about the establishment of fiction (by saying things). The principles speak to this too.
The two places where I see significant reference to
player "narrative control" is in the principles "ask provocative questions and build on the answers" and "sometimes, disclaim decision-making" - one way of doing the latter is to "put it in the players’ hands". But this is all GM-initiated, and to me seems no different from what might be done in most mainstream RPGs. I have used one or both techniques in GMing 4e D&D, Classic Traveller, Cthulhu Dark, Dying Earth, Burning Wheel and Prince Valiant.
The reason I'm going through all this, and honing in on the GM's role in establishing and managing the fiction, is because I want to push back against what, in this thread, I see as an instance of a more general trend on ENWorld. That trend is to characterise any approach to GMing, or to managing and establishing the fiction of a game, that implicitly or overtly eschews railroading, as involving
player narrative control. Which then robs us of a useful vocabulary for labelling mechanics that work in (what Edwards/The Forge calls)
director stance, whereby players wthout GM mediation can directly establish fiction beyond the remit of their PCs' immediate causal influence.
I've read the rules for Fate Core but never played it and am not currently planning to, and so I haven't given them the same degree of thought as I have Apocalypse World. But looking at them right now, I see (p 80) that a player can spent a fate point to (among other things)
declare a story detail based on one of his/her PC's aspect, or to
invoke an aspect. Page 68 says that
invoking an aspect means getting a benefit for your character (or another's - there's an abmiguity in that respect vis-a-vis the option to "pass a +2 benefit to another character's roll").
It seems to me that this does give a player narrative control in the "director stance" sense. For instance, a player whose character has the aspects
Wizard for hire and
Rivals in the Collegia Arcana (this is an example from the book) presumably could spend a point to introduce a detail such as
that the Conclave of the Collegia is meeting tonight. And if a scene includes a "situation aspect" like
Thick mud then presumably a player could spend a fate point to invoke that aspect against an opposed NPC, the logic being that the NPC has become stuck in the mud. As I read the rules, the player wouldn't have to further establish that it is his/her PC who caused the NPC to become stuck in the mud.
Apocalypse World doesn't have these sorts of mechanics. In the "Advanced ****ery" chapter, the following is described as "a pretty interesting custom peripheral move" (p 276)"
When you declare retroactively that you’ve already set something up, roll+sharp. On a 10+, it’s just as you say. On a 7–9, you set it up, yes, but here at the crucial moment the MC can introduce some hitch or delay. On a miss, you set it up, yes, but since then things you don’t know about have seriously changed.
It goes on to say (p 277) "The subtle effect . . . is to expand the player’s, like, area of involvement: into the past . . . [and] move toward a new game, a game based on, but no longer, Apocalypse World."
The absence of this sort of thing from default AW is why I don't think it's all that accurate to descirbe it as a game involving player narrative control.