So, I think this might be a sign of progress, because the "Noble at the gate" story is a great example of goal-and-method. The reasons you say it's not may help us finally get at the disconnect.This, all about this.
The whole goal:method approach is all about the declaration. Whether or not you need to make a roll is based on the declaration. Whether you have advantage/disadvantage on the roll is based on the declaration. It makes the declaration very, very important.
The actual skill of the character only comes up after the declaration, and, even then, only if the declaration triggers a skill roll called for by the DM.
Who judges that declaration? The DM, of course. Which places the DM front and center of all player facing actions. Which means that since the character's abilities don't come up until after that judgment, the character's abilities are less important than the player's ability to make declarations. They have to be.
So, the player uses his Noble background to get past the gate guards. To me, that's not goal:method. That's pretty much purely a character challenge. The player looked at his character sheet which tells him that, as a Noble, he can do this. It's no different than the player casting a spell or making an attack. It's based on the character's abilities, not on the player's abilities.
IOW, there was no "method" being declared here. Any player decisions were made at chargen and not during the challenge.
First, there is no mechanical ability attached to Noble that allows them to influence guards. The mechanical ability that is attached to that background is totally unrelated. So the player is definitely not invoking an "ability" or "skill" when he/she proposes that course of action. He's describing an approach narratively, not mechanically.
Second, goal-and-method or goal-and-approach does not prevent the DM from factoring in the character sheet when deciding if it's a success, a failure, or whether it needs a dice roll. That seems to be a point of misunderstanding. The exact opposite, in fact. Players *should* choose methods that leverage their character sheet, whether that's because of a specific mechanic they have, or because of something related to their background, race, or class that may be relevant, even if there's no mechanic. Goal-and-approach doesn't require support from the character sheet, but it definitely can benefit from it.
Third, how eloquently (or not) the player describes his interaction with the guard is completely irrelevant.
So the only two things that should really factor into the DM's ruling are:
1) The approach, which is invoking noble status to be allowed through the gate
2) The fact that the character doing this actually is (or is at least related to) nobles, which increases the probability of the approach working.
That still doesn't make it an autosuccess. The DM could factor in any number of things, and maybe still require a persuasion check (guards are suspicious that he's really a noble) or, like I suggested earlier, allow that PC to pass but not his companions, etc. But I think however I ended up ruling this, based on the specifics, I would give this approach a higher probability of success than an approach of "I try to persuade the guards that they should let us pass because we have a wood elf in our party, and wood elves are kind and gentle nature-lovers, so obviously we must be nice guys." The noble play just seems to me like an approach that is more likely to succeed.*
And, yes, that is inserting my judgment and biases as a DM into the game. Guilty as charged.
*Now, if they DID choose the "wood elves are nice guys" approach and rolled really well, that would be an excellent time to use roll-then-narrate, and make up some reason why this guard is especially soft on wood elves. Turns out he has been madly in love with the wood elf bartender at his favorite watering hole. Whatever.