Question: If fire annihilates the Dead, why are they such a threat? Wouldn't Jon Snow just ask to borrow Khaleesi's babies for an afternoon?

I'm aware that Dany is the only one who can control her dragons. 

My point is this: Jon Snow is trying to unite houses, tribes, and kingdoms the world over against a common enemy, continuously reminding all that will listen that they have never been up against a force so formidable as the White Walkers and the Army of the Dead, and that only he has seen what the Night King can do, and that if the world of the living is to endure, we must rally every warm body against them. 

But I fail to see the magnitude of such a threat in a world where one of its residents can control a weapon that administers a method of death (on a semi-mass-destructive scale) that happens to be the exact bane of such a potential threat. 

Why is Jon Snow so concerned with boots on the ground and global cooperation when it seems that one couldn't come up with a better antidote to the walking dead than dragon-fire?

Answer: 
That's essentially why Jon is there.

When Jon received Tyrion's letter inviting him to Dragonstone, Davos Seaworth pointed out to Jon that fire kills the Others and asked him what breathes fire?

From Season 7 Episode 2:

> 
> Sansa: You think it's really Tyrion? It could be someone trying to
>  lure you into a trap.
> 
> 
> Jon: Read the last bit. "All dwarves are bastards in their father's
>  eyes." 
> 
> 
> Sansa: What does that mean? 
> 
> 
> Jon: It's something he said to me the first night we met. You know him
>  better than any of us. What do you think? 
> 
> 
> Sansa: Tyrion's not like the other Lannisters. He was always kind to
>  me, but it's too great a risk.
> 
> 
> Jon: "The Seven Kingdoms will bleed as long as Cersei sits on the Iron
>  Throne. Join us. Together, we can end her tyranny." 
> 
> 
> Davos: Sounds like a charmer. Of course, the casual mention of a
>  Dothraki horde, a legion of Unsullied, and three dragons a bit less
>  charming.
> 
> 
> Jon: What? 
> 
> 
> Davos: **Fire kills wights, you told me. What breathes fire?** 
> 
> 
> Sansa: You're not suggesting Jon meet with her?
> 
> 
> 

So that's why Jon is there. He wants Daenerys to aid him in his fight with her dragons. He went against his Lords and Sansa's counsel and went to Dragonstone so that he could persuade Daenerys to send dragons for the Battle for Dawn Part II.

He can't borrow them as such, as he can't control them (As of Episode 5 Season 6, since this looks like I am implying he won't ever be able to control them). But he can ask the person who can control them (Or one of them at any rate) to come with him.
Giant Wights with Large CrossBows like the one seen in S7E4-can slay a dragon. The top of the wall was a lot farther than Bronn was to the dragon with the Scorpion-shooting dragons with large bolts wouldn't be too hard for a giant.