What do Jon Snow and the party really hope to gain?

Question

In Season 7 of Game of Thrones, episodes 5 and 6, we see:

Jon going north beyond the Wall, on a mission to secure a 'live' wight they can then bring back to Kings' Landing.

It's really not clear to me how this mission had any merit to begin with:

1. Even if they succeed, there's no reason for anyone in Kings' Landing to believe that there is more than one such specimen, or that White Walkers exist, or that the wall won't stop them.

2. Even if the south is convinced that the army of the dead poses a threat to them, It seems unlikely that Cersei will suddenly abdicate. It's not as if she has the common people's support now, so it's unclear what is to be gained.

3. Jon already knows that wights don't travel alone, does he really think it's likely he'll just pick one up and escape without some miraculous outside assistance?

All in all, the entire plot seems rather hare-brained at best:

What's the in-universe reason for all characters simply going along with it?

Answer 1

1. You have to remember at this point everyone south of the all thinks the undead are a myth - bringing one would help shatter that illusion. Imagine someone coming to you, claiming that there is an army of the undead marching south. You don't think the undead are real - do you believe it? Now, they show up with one wight and further make this claim; suddenly you find yourself much more inclined to listen.

2. Jon's goal of going to King's Landing with a wight isn't to have Cersei abdicate, but rather negotiate a cessation of hostilities so they may join forces to face a more grievous threat. Once the Night King and his army are dispatched, they can resume their petty bickering.

3. As for why everyone goes along with Jon's idea, Jon is King of the North. When your king decides to do something, you do it.

Answer 2

Granted, it's a silly idea, Dany even says so to Tyrion in the Beyond the Wall episode, yet the point of doing this seems to be:

It's not just a dead-man-walking, it's a dead-man-walking from Beyond the Wall

The whole point is to show to people that there is a real threat from the North, and not just to the North. Jon is trying to inspire people to realise that their very lives are at stake, not just their homes or moneys. The Night King will come for them all; he's an enemy of life itself.

This one's important, and it reconciles the fact that out-of-universe, we know Cersei already has unGregor (undead Gregor), and in fact, there may be some in-universe who realise this as well. So showing the undead from the North, will prove to people that the threat exists, and it adds to his and Dany's credibility.

It's not just for Cersei, it's for the Kingdom

By showing the people of the kingdom that they are protecting them, when Cersei is merely hoarding power will get the population to truly believe in Daenerys.

He's essentially doing what Stannis thought; to:

save the kingdom to win the throne.
_{-A Song of Ice and Fire: Book Three: A Storm of Swords, Chapter Seventy-Six (Jon XI).}

It's not just for the Kingdom, it's for Dany

Jon is there to prove to Dany, not just the threat, but his loyalty too. He's willing to risk his life to show that this isn't a joke. Their petty squabbles no matter how serious or important, are nothing compared to the War for Dawn.

Answer 3

Actually, the whole expedition was almost useless on its face value, it was silly and dangerous but has had many side effects. The first is that now Danny believes Jon and now they will fight together with a clear understanding of the most urgent threats. The second most important result of this expedition is that the Undead King now has an undead dragon. Storywise, this means Danny is vulnerable again. Finally, the living have realized that is no way to destroy the undead unless the Undead King is destroyed. He will keep on raising more dead otherwise. The good thing is destroying the UK is probably going to destroy the whole undead army in one go.