There are two things here:
(1) If someone has failed, generally the GM is going to make what in AW parlance would be called a hard move. Some implicit or perhaps expressly flagged consequence is brought home. Burning Wheel is probably a bit less clear-cut here than AW, in part because it is based around scenes and stakes for resolution (AW is not in the same way), and in part because it has extremely high rates of failed tests (by D&D standards, and setting aside low level human thieves in classic versions of the game).
Overall I like this post. Let me try to bring up a few of the things people just aren't getting.
1. I think there is some misunderstanding about what you mean by implicit. For us, the idea of kitchen in the house is implicit and that attempting to break into the house at an unspecified time has an implicit risk of a cook being present. Maybe that's not exactly what you mean by implicit? Or maybe time of day in these games is necessarily more codified than we imagine?
2. Part of what's happening in our discussion is that the scene is subtly being changed midstream as we walk through this example, from 'house at unspecified time' to 'house at 2am'. I think we all agree that a cook in the kitchen at 2am would not be implicit, which would seem to align with what you are trying to say above. I think this answers the cook in the kitchen at 2am critique very well as it just wouldn't be something that will occur at least without a prior soft move establishing the cooks presence.
Though this still leaves open the scenario of
A) a soft move previously established the presence of a cook in the kitchen at 2am
B) and possibly others
(2) A general principle in AW is that moves have to follow from the fiction. Burning Wheel doesn't state this (and doesn't use the terminology of moves); it talks about scenes, but again takes as given that the elements in scenes are implicit if not express. (That's part of the explanation of why a player can make a check to see if there is a chamber pot or jug in the sick room, but not a check to see if it contains the Imperial Throne.)
The notion being that chamber pots and jugs are typical to sick rooms and thus implicit to them. Similar to the way a cook is implicit to a kitchen but not exactly the same.
But the last part of this about a player making a check to see if those things are in the room, I really want to walk through that as it aligns almost perfectly with my Scenario A above.
Scenario A (expanded) Assuming our premise is breaking in at 2am, the player checks the house to ensure no one is awake (or maybe that's not a valid player move in these games?). The player fails the roll such that a soft move is called (scenario A) and this establishes the presences of the cook in the kitchen. What the player does next would have the possibility of triggering a hard move involving the cook screaming for help. To make this more concrete let's say the player threatens the cook, 'if you scream i'll kill you', which triggers a hard move where on failure the cook look at his rather large kitchen knife and says just try it while he yells for Help.
But there's a couple of things with this
1a - It's fairly clear that instantiating the cook screaming for help over a soft move then hard due to the where and when being somewhat implausible otherwise clearly makes for better gameplay doing it over just a hard move in this scenario. But, is handling that over soft then hard move actually changing the conceptual critique any? I'm not really sure it is.
1b - What is the game incentive for the player checking for anyone awake in the house? It seems like the fewer precautionary moves you make the fewer potential obstacles will get put in your path? I presume not making those precautionary moves will typically influence either your failure rate or strength of your effect on some future moves, but it's not clear whether that's necessarily the case and whether the player is informed of those potentialities beforehand. This is cutting to both an 'actually informed' critique and a 'gamification' critique.
Where is your mooted cook coming from? Your mooted kitchen? The lack of expectation of the door being passed through?
Implicit...