PHB page 175, variant rule: Skills with Different Abilities
A Wizard wants to move a rock, I might give him an INT check and he figures out how to use a lever. He wants to open a sticky door, he could use INT to figure out to tie a rock to a rope and swing it like a wrecking ball. The Cleric could use WIS to figure out the same thing because it's pretty much common sense to use a machine when you don't have the strength to do something manually.
It is something of an aside, but what is the conversation here? Does it start like this?
Player(Wizard): I try to move the rock.
DM: Make an INT check.
If so, for me that is problematic in a different way.
Not every check has to be an intuitively obvious choice of ability to use.
Um, yes and no? The player should describe a goal and an approach. If the DM thinks that the success of the approach is uncertain, then the DM calls for a check based on the ability (and possibly skill) she judges most important in determining success of the given
approach. I would say that the ability(skill) is
usually obvious from the approach, but there are certainly exceptions.
This has on occasion been pushed by some of my players with high DEX and low STR score PCs that want to use some type of parkour to climb.
That seems just fine (as long as parkour is appropriate to the circumstance, which it often isn't even though people desperately want it to be).
Not every game result has to be narratively explained based on what mechanics are in play, and not every task has to use the same game mechanics.
I agree with the second part, provided that by "task" you mean "objective" or "goal". There can be different approaches to achieving the same goal. However, a particular approach will generally have a particular mechanic associated with it. And in your examples you do seem to be following a methodology that associates narrative and mechanic "appropriately".
But then in the first part of that summary statement, you turn around and say you are ok with dissociating the narrative from the mechanic.
I think perhaps dragging in skill checks as examples was a poor choice on my part because they generally start with a narrative element - the player's action declaration, whereas a saving throw starts with a mechanical element.
IIUC, your basic point is that one can imagine more ways to mitigate damage from a fireball than just jumping out of the way (or the other things that are generally seen as DEX related). I agree with that, but I just don't see much value in having those as possible narrative elements without appropriate mechanics. And I see narrative that is dissociated from mechanics as a big drawback. I mean, narrative is constrained by mechanics all over the place. In fact, it seems like one of the 'pillars' of DMing is weaving a narrative that is consistent with the mechanics (even though not totally determined by them).
But now (because I love arguing with myself), let me guess that your point would be that your narration
is consistent with the mechanics because the only thing that gets recorded in the mechanical game state is that the target took 1/2 damage (or whatever); the fact that it was a DEX save actually has no mechanical consequences.* Thinking about this, the only similar situation that comes to mind immediately is damage type - damage type might affect the amount of damage taken, but once that is determined, the type of damage usually makes no difference in the mechanical game state. Despite this, I would never narrate acid damage as something that looked like bludgeoning damage. Would you?
OTOH, maybe you have a different reason for thinking that a somewhat dissociated narrative is ok, so I'll stop putting words in your mouth. And I apologize for the fact that 'dissociated' has a pejorative connotation - I'm not trying to be snarky.
* This is beginning to remind me of those diagrams, the name for which I cannot remember, about how things bounce back and forth between fiction and game state. And how abstruse the discussions around them became such that they ended up being useless even though at the beginning it seemed like a useful idea. But I digress.