Did you see a scene where a White Walker was hit directly by dragon fire and survived? It looked like the dragons were focused on defending the heroes surrounded on that rock outcropping, while the Night King and his officers were watching the action well out of range of the battle.
What we have seen was the Night King and his lieutenants crossing the ring of fire created by the Children of the Forest outside the old Three-Eyed Raven’s tree. But it looked like his frozen aura put out the flames as he crossed the ring.
As far as the question “why Bran?” I don’t have a theory, but I think there is a literary precedent for why he ends up being the One.
In almost every classic story, only people who are deeply wounded can become great leaders. Bran is crippled before his powers begin to manifest. Without the pain and humility brought about by his accident, he is a stupid little boy who is undisciplined and unwilling to work hard at his training, preferring to break his promise and climb the castle walls.
In a similar way, Jon becomes a great leader even though he has no desire for power, and only becomes a king after he is killed and resurrected. The wounds he carries (the loss of his family, the loss of his first true love, and survivor’s guilt) all keep him humble and focused on being a servant even though he is hailed a King of the North.
If you look at the other main Stark children, they all carried too much arrogance, anger or fear to be transformed by their wounds (aat least up to present): Robb rushed off to avenge his father, then thought he could break his oaths during the war; Rickon was too young to handle the visions he shared with Bran about their father’s death and was driven slightly insane; Arya became a serial killer, bent on finishing her list; and Sansa became a gifted player of the game, using misdirection and political scheming as her weapons of choice.
Who’s to say that the other children couldn’t have developed these powers under the right circumstances?
Bran reminds me a little of Proffesor X. Maybe there’s some archetypal thing going on here.