This misses the point of the example. I thought everyone at this point was aware that the problem situation at hand require the cook to be there
because failure. So, yes, it is an underlying assumption that we know what would happen on a success. So if your "you don't" claim that the example do not specify the successfully check, that is you misunderstanding the context.
If you "you don't" refer to that the player in the hypotetical game wouldn't know that is also an interesting but ultimately not relevant point. It is true this would not be a
problem for the player if they are left in the dark, but that is not guaranteed. The mechanism for how the player get to know about it is unspecified, but it is clearly not impossible. For instance the GM should have a pretty good idea of what we planned to do on a success, and they might "confess" this to the players. Maybe more likely might be the player sensing a certain pattern emerging based on what happen when they succeed and fail rolls.
I just want to second
@Maxperson reply to this,
how things is established in play is important. Even the GM taking inspiration from skill checks when determining content carries the risk of producing weird correlations that over time can build up to something hard for the brain to accept. The brain is
very good at detecting patterns.