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: 
As an author, your primary task is to create *conflict*, *not* solutions. When you design a story, you always try to find the most complex and perplexing way to resolve a situation.

To put it another way: You can't fill a series of 700+ page books when you're solving every problem efficiently.

In this case, dragons are a natural/efficient solution for the Walker problem, so George has to find obstacles. In game design this is called "balancing". Games are no fun when one party is so powerful that they can simply wipe out the opposition with a click of a button.

Here is a short list of complications that I came up with in a few minutes:

* Advisers playing their own games
* Misunderstandings
* Traitors
* When Daenerys arrives with a few or one dragon, they might be killed or, worse, injured (dying for days or weeks! The Horror!) So she should arrive with many - which she can't really control. And while she's there with many dragons ... why not take the throne while we're at it?
* It's not good for your reputation score when your dragons burn a few cities and eat the people there on the way ...
* Daenerys arrives with many dragons, defeats the Walkers. Now everyone else thinks "hey, she's weak now, let's finish her!"
* She returns home only to find that someone attacked while she was away
* How long can dragons live in cold conditions?
* How efficient is magic against dragons?
* If it was so simple, why isn't everyone asking Daenerys for help? Let's ponder this to see what else we can come up with.
* Maybe the Night King can somehow take control of a dragon
* Or interrupt Daenerys control while she is riding (turning her into an in-flight snack)
* Maybe the dragons don't want to. They are somewhat sentient, right? They can (and might) say "No". Good luck forcing them...

I think it's George's great skill that he can come up with so many believable complications and problems in his story.
No. He (or any one else for that matter) can't ride, control, or order a dragon to do something that complicated. Even Daenerys can't. 

During the series, you never see her riding any of the other two dragons. They're not with her when she attacks the Lannister army because (most likely) she can't just tell them to follow her.