It's all so silly and hurts Jon as a character too. It bothers me a lot how reckless the idea is and how it's reliant on characters being too stubborn.
Didnt he just receive a letter from Bran, who he didnt even know was back, and told him he "saw" the walkers at Eastwatch?
And Jon just immediately decides its go time.
Dany could have sent a dragon if her whole deal was seeing to believe. I dont even know why they are even trying with Cersei. If it's a matter of showing the Walker to the world...why would it matter if peasants at King Landing knew of them?
The Maesters seem to be a big deal, so why not take one of them to see the walkers instead of bringing one to Cersei of all people?
I love the atmosphere beyond the wall, but there's just not a good enough reason for it. Add to the fact that it's a suicide mission, with no promise it will work.
Maybe if the dragon glass was there, or if they wrote in some kind of plot item like a much needed ice rock that was necessary to mine the dragon glass or something like that.
Dany does not fully believe it and the other problem is if she goes and does see that the threat is real then she has an obligation to help but she doesn't want to do that until Jon acknowledges her as the Queen of Westeros and denounces his title as King.
Also the problem with sending just a dragon alone is that Dany can't talk to them so she would need to ride on one to go and see and that alone is a pretty big risk if something were to happen to her.
And bringing the wight back isn't to show the people of King's Landing, it is to show Cersei so that she can see the threat too and then the plan is for everyone to "join" together. Of course that is not going to happen as Cersei will most likely try to backstab Dany or Tyrion but that is Jon's whole plan for this.