As an outside observer, the player introducing 'the Falcon's Claw' doesn't seem to really be related to the Beliefs, or if so only in the most tangential of ways.
Which Beliefs?
I would have to go back to old notes to see if I even have a record of the PC Beliefs at that point in play. So I don't quite see how you are across them!
At one stage there was a Belief about finding magic items that would help free the brother from possession by a Balrog. I don't recall to what extent that Belief had changed/developed.
it kind of seems the Falcon's Claw was established in the fiction more by player fiat than anything else.
In RPGing, I know of two ways to introduce new content: roll on a table, or have someone make something up.
Burning Wheel does not use rolling on a table. As per the quotes from the rulebook that I set out upthread, framing and consequences are deliberate, and intended to challenge player priorities and give expression to the consequences of those challenges.
So someone has to make it up.
Sometimes the GM makes it up, and incorporates it into framing, or (as the black arrows illustrate) into a consequence. But that is not the only way. The point of a mechanic like Circles, or Wises, or Scavenging - looking for people (or hoping to meet them), recalling stuff, looking for stuff - is to create a framework within which player priorities for the fiction can be given effect to.
As a
player, I made a Scavenging check when Thurgon searched Evard's tower for spellbooks. At the table, who invented the idea of Evard's tower? Me, the player. How did it come into play? Via a successful Great Masters-wise check for Aramina. Who invented the idea that there would be spellbooks in the tower? Me, as part and parcel of making the Wises check in the first place (Aramina had a Belief about finding spellbooks). Who initiated the Scavenging check? Me, playing Thurgon (Aramina was Taxed to unconsciousness from an attempt to cast a spell).
And when the Scavenging check failed, who invented the letters from Xanthippe to Evard, that tended to imply that Thurgon is the grandson of a demon-summoning wizard? The GM, introducing a complication that challenges Thurgon's Beliefs and Relationships. (Very analogously to the black arrows.)
This is how the game works.
Now, upthread
@Citizen Mane mentioned the passages from the rulebook that speak about "no beam weaponry in the Duke's toilet" and "no superior gear-mongering in the village". What these really are, in my view, are statements of the basic principle of the credibility check. Robin Laws says something similar in the HeroQuest revised rulebook: if the genre of the game is a western, the fact that the cowboy has a descriptor Fast 16 while the horse has Gallop 14 doesn't mean that the cowboy can outrun the horse.
Generally, if everyone is on the same page as to credibility the issue doesn't come up. If there is some uncertainty over what's credible, it can be worked out via conversation among participants. If, despite such conversation, the most interesting action declaration that a player can think of is a Scavenging check to find an unmotivated vorpal sword in a random cupboard, then - to again echo Citizen Mane - the game has gone so badly wrong that we don't need rules to shut down player agency. We need all the participants to revisit the basics - why do the PCs have no priorities? why can the GM not frame interesting scenes? why does play seem to be unfolding in complete disregard of possible consequences?
Once those issues of ethos and expectation are resolved on the part of all participants, then the allocation of roles will do its job: the GM makes stuff up as part of framing and consequence; the players introduce possibilities as part of the declaration of their Circles, Wises, Scavenging, Perception etc tests, and if those tests succeed then the possibilities are realised.
This is not the only approach to high player agency RPGing - Apocalypse World exhibits a different one - but in my experience it works pretty well.