I think 5E would benefit from an attempt to incorporate "Let it Ride" and explicit intent and stakes into its resolution mechanics.
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Given that D&D is primarily built on task resolution instead of conflict resolution, and generally has a bigger need for the DM to keep secrets, it might also work out that the incorporation is no more than a section of advice and guidelines. But even that would be superior to not addressing the subject at all.
One of the Save My Game columns last year suggested using Let It Ride (not under that name, but the same mechanic - ie no re-checks).
Once you incorporate that sort of rule, I think something has to be done about stakes - otherwise it risks becoming just a roadblock to further progress by the PCs, or an excuse for mere GM fiat (in the bad sense of that notion).
The 4e DMG, and even moreso the DMG2, are full of stuff about "failing a skill challenge isn't the end of the adventure" (and maybe similar stuff for combat? I can't remember), but don't actually explain how to make that so.
And I think it's non-trivial. For example, techniques that a game like BW uses to keep paths open includ giving players a degree of narrative control (eg via Wises, vial Relationships, via Circles) and giving them a strong role to play in setting the priorities for a scenario (via Beliefs).
Once these things are stripped away (and that is part and parcel of the "GM secret" aspect of D&D that you refer to) new techniques are going to be needed. And it would be good if they could be more focused, and more workable, than just "Build your world and let the players sandbox their way through it."
Addressing this issue can also address the issue of players feeling more confident to risk failing, and therefore to not always try and push their best scores. And one might hope that addressing this issue would also involve linking the fiction tightly into the adjudication of action resolution, which also might open up a wider space for the players to be comfortable trying stuff.
A simple, maybe half-baked example: if the player is confident that a mug-crushing attempt at intimidation, even if it fails, will open up one sort of alternative path (say, someone who sees the PC humiliate him-/herself and takes pity), and that a silver-tongued attempt at trickery will, even if it fails, open up a different sort of alternative path (say, a member of the guild that the PC pretended to belong to notices the attempt and starts following the PC), then the context for choice becomes richer. The stakes (within the fiction) become more varied. The player has a more complex circumstance for choosing than simply "What is my best stat".
But setting up these sorts of situations, as a GM, requires advice and support. And if the support is not going to come from the players (as at least some of it does in BW), it's going to have to come from somewhere else. And in a way that gives the players confidence that the options are there, even though their character sheets aren't point them to them (as they might in BW or a similar game).