I haven't played Burning Wheel yet, so my knowledge of it boils down to having read through the book a bit, and then mostly your accounts of play. So I know it is different than how Blades in the Dark functions.
I can see the idea of collaborative storytelling being used to describe Blades. I think very often that gets overstated....it's certainly nothing on par with games like Fiasco or Microscope, but there are elements of it in Blades. So I think I get it as a descriptor, but I think its importance is overstated. It is a small part of the game. I think it's also a part of most games, including D&D, but it's more present in Blades.
However, there are players out there for whom ANY instance of player decision that is not 100% "in character" is a deal breaker. Or at least, any such decision that isn't somehow justified through exception of some sort. For such folks, the idea that players in Blades can select or influence elements of the Score, for example, doesn't work for them.