Critical Role ...
I also don't think that D&D is especially relevant in this thread (which is in the General forum). For one thing the OP is not a D&D player, but rather prefers Savage Worlds and is curious about Burning Wheel.
Given that the vast majority of people in the hobby are playing D&D (mostly 5e) and thus - if they're reading this - will be trying to fit what's said into that paradigm, then D&D is always relevant.
Secondly, D&D - at least in its 5e variant - has many mechanical limitations that get in the way of mechanically-driven character arc play; namely, it has no mechanics for player-imposed finality of resolution outside of combat.
Doesn't make sense - you say it has mechanical limitations and then your example is a situation where is has no mechanics?
'No mechanics' is never a limitation; and in this case 'finality of resolution' doesn't sound like a good idea. In combat, yes, there's usually a finality - one side or combatant dies, or surrenders, or is made captive, whatever; the combat's done and in most cases it only has to be done once.
But in a character-based or social situation, unless a combat-worthy finality is somehow applied (e.g. one character charms or captures another) the situation is never truly finalized, and thus applying a finality mechanic seems...odd somehow.
For example, take some of the various discussions we've had on here in the past. None of those threads have to my knowledge been shut down by the mods (analagous to the DM applying a finality mechanic) which means they're all open to rebooting and review in the future - they're not final (despite our best attempts!).
Same is true of a diplomacy attempt - unlike a combat which when it's over it's done, a diplomacy attempt or in-character discussion (or argument!) is never really over as long as all participants remain in control of their own thoughts. People can have second thoughts, or realize they erred, or come up with alternative arguments, whatever; and applying finality mechanics just doesn't make sense from a pure roleplay point of view.
D&D is guilty of this with its diplomacy family of mechanics over various editions (yes, even all the way back to henches' morale and loyalty rules) - the mechanics insert themselves to try and finalize what would otherwise be ongoing and perhaps repeating roleplay.
As I've already posted a couple of times, I don't think that those particular sorts of bells-and-whistles are crucial for what the OP is looking for. But there are certain things that are crucial. The most important of these is dropping the conceit of "the adventure" or "the story" - and replacing it with character-centred framing on the GM's part and openness to action declarations and resolution outcomes on everyone's part.
It's not an either-or.
A character-driven arc - be it for an individual character or several interacting - can quite happily weave its way through and around a background or setting-based story and-or a series of adventures
provided the DM is willing to allow the table-time for the character arc(s) to play out when required.
And this, I think, is where things fall apart at some tables: the DM (or some players) aren't willing to spend that time. They're not willing to allow
@Sadras ' sibling PCs to spend an hour or so developing their arc, or my PC and a few others time to argue over how we intend to approach our post-adventuring political careers, or the Cavalier PCs time to participate in a jousting tournament (with all the attending social affairs) where results both social and field may affect their reputations.
In the three examples I give here the only place game mechanics need to appear at all would be for the Cavaliers, both to determine their jousting results (a variant on combat mechanics) and perhaps to determine any lasting effects on their reputations afterwards. Otherwise, it's all roleplay - so let it happen!
