are you serious? This has literally been the apocalyptic doom threat from the very first few minutes of the show/book before the opening credits ever came into existence. It's been a major plot point literally from the next scene after that that no one believes these things exist ("A mad man sees what he sees"). Bringing back one of these things fully intact for people to look at with their own eyes, see it raging and rattling in a cage like a feral meth'd up baboon has a powerful dramatic and emotional effect far beyond the logical argument that "Something is clearly up there that is disturbing the Northerners and is worth checking out" and would make a whole lot of people re-think being complacent.
As well, the obvious subtext of the whole plan, which I feel you seem to have missed, is that getting this intact wight isn't about convincing Cersei, it's about convincing Daenerys. Despite her obliging him his stupid plan, it's pretty obvious that even if its best case scenario and they all return with a bit of frostbite and no Wight, she'll pat them on the head, said "You tried", having never believed him to begin with and having lost little of value in letting him try, and go back to dealing with Cersei.
Westeros is (ostensibly) set in an era of perpetual ignorance and superstition. Very few people are literate, and even those that are literate likely aren't very well educated in things like the sciences, logic, rhetoric, psychology, and sociology. Giving them a series of explanations the way Jon has been doing, amounting to a Pascal's Wager of "If you help us, and they don't exist, no big deal! If you don't help us and they DO exist, we're all dead!" isn't going to sway many people, especially in the middle of a war.
These people need a dramatic exhibition. Bringing back a Wight, as rotted as possible, clearly standing on its own, of its own volition, making odd shrieking noises, clearly able to handle a weapon, running jumping howling slashing, trying to escape from its cage or what ever it's restrained in, is going to make a HUGE impact on people. They're going to look at that thing and, even the most cynical, would think "If this is a magic trick or some kind of necro-puppet, that is a whole lot of effort made to persuade me of something I didn't think exist is trying to kill me, maybe it's worth at the very least sending some reconnaissance people I trust to go up there and see for themselves"
Cersei's rule and remaining power is based on fear and money. In near future her army will consist of mercenaries form Essos who won't give a about the army of the dead.
They will give a when they're being slaughtered by the army of the dead.
For Cercei's subjects I am not sure whether they are more afraid of the dead or of a dragon queen who is burning whole armies and people who won't bend the knee.
They'd be more afraid of the DEAD, because the Dragon Queen can be stopped very easily by just surrendering! Pushed to the absolute brink, I can guarantee there would be half a dozen or more Kingsguards and more Goldcloaks who would willingly bring Daenerys Cersei's head if she brought her Dragons up to the red keep and just had one of them knock with its crinkled little wing-hand.
Jon and Davos should know that they can't stay away from Winterfell for too long and expect everything to stay in order. Further more Jon has to lead the North against the Night King which he certainly knows. Gendry would be much more valuable if they used him to get the allegiance of some remaining houses of the Stormlands because everyone should be able to see he is Robert's son.
Why wouldn't order stay? Sansa is a Stark, and she's family to Jon. Jon doesn't care about being a king or a lord, he only cares about the safety and well-being of the people. If the Northerners did a soft coup and told Jon they've made Sansa or Bran the new King in the North, but they're still going to help fight the White Walkers, he'd say "Okay" and get back to work.
And again, no one knows who Gendry is, and they don't know for certain that revealing him would make any difference. As far as we know, the Stormlands are wasted and not able to field a significant military force. And considering Cersei considers the North a threat, it means she plans on invading them, but she is being held up by the threat of Daenerys. If Gendry's existence comes about to her, she will straight up deploy Euron Greyjoy up to White Harbor or up the western coast to just rape and burn the out of everything and everyone, regardless of Daenerys, out of sheer paranoid rage and anger. She might even crack down on the Stormlands pre-emptively. It might help thin out her army for Daenerys, but it would result in a whole lot more bloodshed overall.
In the moment they are all acting somewhat unreasonable in my opinion. I just don't think this is really Game of Thrones like and I am pretty sure that GRRM would never write something like that.
I think this is exactly what he would do. In fact, he already did something similar in book 1 with Alliser Thorne bringing a Wight hand to the king, but the hand was dead and rotted and unmoving.
This overall plan, viewing the Whites as the biggest threat, is one hell of a bargain. If it works, the entire realm wakes up to the reality of the threat and takes measures to addressing it. If it fails, who is lost? Some low-level people, rebel knights, a Night's Watch defector, not any important lords or counts or knights or anyone of great power or wealth or influence.