But there must be someone adjudicating the world, right? If the intent is to find a ship captain and they succeed, who describes the NPC? Who decides how large the ship is and what the fee will be?
I appreciate your description of the mechanics you had in mind which followed. But I think it is getting into the weeds of system, and doesn't for me explain the salient point: why do you think the fixed world sandbox can be railroading, but player driven is not? I don't think this requires a detailed explanation of how tests are made to be answered.
I also appreciate your attempt to bring nuance to these scenarios. But again I fear that it is over complicating the analysis and I still don't have a good understanding of what precisely you think is railroading. At the moment it seems to me like a "I know it when I see it" standard. And maybe that is the extent of it.
If that isn't the extent of it, what is the minimal working example to your mind that illustrates it? Can it be captured in a sentence or two? E.g., "If the party wants to head east but the GM comes up with unrealistic barriers which prevent their action then it is railroading".
I feel I've seen a lot about specific mechanics, but little about the general principals that make them better or worse (in your view) for these purposes.
I think I understand Burning Wheel more from the Wikipedia entry The Burning Wheel - Wikipedia than from the descriptions we've seen. I'm still unsure how you would weave any of this into a long term campaign of any kind, I'd have to watch a live stream or similar to get a better idea.
It just seems to have more player directed play than anything. I don't think it's necessary for a sandbox, it's just a different approach.