You're kidding, right?
The fact that many of us have said that it was a bad example, and yet you keep bringing it up as if it's the way to do fail forward, is a ginormous strawman. Even though we have shown you many other examples.
The definition of strawman "An
informal fallacy of refuting an argument different from the one actually under discussion, while not recognizing or acknowledging the distinction." I never said it was a good example. I brought it up as an example that I had found so that we could discuss a detailed example in D&D instead of talking about how it works in other games that have different approaches and assumptions.
Stop sea-lioning me on this.
I pointed this bit out earlier, although I will admit I forget if it was to you or to someone else.
When someone rolls badly on Stealth, there are no rules that say why they failed. There's no d% table of Reasons Why You Failed The Stealth Check. Yet you don't just expect the character to shout out "I ROLLED A FOUR!" a la OOTS, and I'm sure you don't just shrug and say "you failed" and that's it. Instead, I'll bet that either you or the player make something up. You make up a reason why the Stealth roll failed. Right?
The fail forward/partial success examples we give are exactly like this.
No, the rules don't say that there's a cook that hears you. But it makes sense to say that because you bungled your roll, you were clumsy with your attempt to pick the lock, and because you were clumsy, you made noise and were heard, or you didn't keep to the shadows enough and were seen. Just like it makes sense to say that you stepped on a twig or knocked into some furniture and that's why you failed your Stealth check.
Remember that fail forward is there to keep the game moving. So such an event should mean that either another way through is revealed--someone ages ago brought up the idea of "you can't pick the lock, but you suddenly notice an open third-story window"--or they succeed but there's a cost--such as the screaming cook, or cutting yourself on the jagged lock, or damaging or losing your tools.
It does fit if there's a time crunch involved. If you only have an hour to get in and get out, then spending ten minutes on the lock will mess up the rest of your plans.
The issue that people have is that it assumes that there is a cook active and in the vicinity at all hours of the day. As a DM I set up interesting obstacles and scenarios when planning. I am not being particularly neutral in my judgment here. I'm thinking about how I'm going to handle the boring bits using minimal game time, if it makes sense in-world for there to be a threat to great for them to handle how do I broadcast that they should avoid it and how, and finally the bits and pieces we're going to play out. So those bits and pieces we're going to play out should be challenging, interesting, and just as important not be gated behind a single roll so I need to consider multiple options. If I think there even is a cook, I may make some notes about servants quarters and whatnot such how they're going to react (not all servants are going to be fond of their employer and so on) to intruders.
As much as possible once the session starts I become a neutral arbiter and referee. I break down what's going on in the world into reactions to what the players say and what actions they attempt. I am not pushing a narrative, I feel no obligation to force the momentum of the game in any specific direction. The players are in the driver's seat not me. If they try something I had not thought of ahead of time I'll have to make a decision at that point, if I'm uncertain (such as whether there a coal chute or root cellar) I'll roll for it. For me, the way forward if they can't open a door is to be attempt to open a window. If they can't open the window there's a brick they can use to break the window or whatever other option makes sense for the scenario. Of course there's going to be times when I'm ad-libbing because the players went in a direction I had not expected and I'm making it up as I go along. But I'm still going to do that ad-libbing based on what I know about the location and who or what they might encounter. The location of the cook? They will only be there if I think they would be preparing food or cleaning up at the time of the break-in.
There are some situations where I can see fail forward could apply, although most of those for me I would consider partial success or success at a cost. So the player rolls high enough to break down the door but takes a bit of damage doing it. But there's still the chance that they could just not be able to break down the door. For example I don't know how I'd fail forward if the player needed to make a knowledge check to a mystery.
There are other cases where, like with lock picking, the sleight of hand check was done to see if the character could pick the lock quickly and they may fail. If I think they are capable of opening the lock if given enough time they have the option of rolling to see how long it will take, but that extra time increases the chance of being noticed. The big difference to me is that I see those as separate action declarations made by the player, their declared action didn't work so they have multiple other possibilities including fiddling with the lock until it does work.
I understand the concept of fail forward and in games that don't have the same core assumptions of the effect being directly tied to the cause that D&D does, it would be easier to implement. If I was playing a game where failure could mean bad karma, that bad karma could have influence other than a lock just stubbornly remaining locked. I am someone who needs concrete examples sometimes, theory is great but the only thing that really matters is the specifics when the rubber meets the road. So if you want to discuss actual detailed examples using the assumptions of D&D? Great. If not, there's no reason to continue this tangent.