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: 
I think Jon is being practical. Yes the dragons are great but the night king has magic and could easily blind them in clouds of snow. The army usually walks under a blinding blizzard so it isn't a sure thing.

Plus the biggest issue he has to deal with south of the wall is everyone killing each other off or worse attacking him from behind when they are trying to defend the wall. He has to unite everyone to stop the bloodshed. 

Another thing to think about is that the night king may very well drag all those dead bodies back out of the ground to join his army once he is past the wall. The problem is no one knows how great the danger is so getting everyone ready and united is really the best option.

--- edit ---

After S07E06 we see exactly why the dragons aren't the perfect weapon verse them. I won't expand upon that but another thing I realized is that in that sea of undead, which includes giants and polar bears the actual white walkers are not going to be easy to see. The Night King and his group riding horses seem to just appear on the battlefield. They may remain hidden until they are ready to act.
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.