I think the Brotherhood is the only group involved with any sort of ambiguity behind their actions, from their perspective not from ours as viewers. Jon, Dany, and Cersei's POV and current plans all make sense from their own perspectives though.
The only chance Jon has of convincing the southerners of the white walkers is to show them one, so that's what he's doing. He's not one to delegate and he's already faced them so it makes sense he's going. Everything with the people he brought makes sense, a limited group he trusts to grab one and go. Thoros, Beric, and the Hound though? Their motives aren't clear but they're along as redshirts, basically. Someone has to die so why not introduce characters that people know and don't factor into larger plans or stories to be killed off.
Dany, and Jon, are doing what Tyrion said though, getting proof for Cersei, and Jaime because he can actually assess the threat and help sway Cersei if it's legit. Jon doesn't know Cersei or the Lannister's, while his family and the North fought them he was beyond the Wall with Wildlings. Dany trust Tyrion to a point and she's risking nothing, all she has to do is wait for Jon's return and if he brings one with then she knows she needs to head north to battle the army of undead before they reach her and her army. That's part of why Jon is doing it also, Dany said she'd help him if it's legit. With her Dothraki, Unsullied, and everyone else in addition to the armies Jon has, that might be enough to defeat them. Dany has to hold off on attacking Kings Landing but she knows she can wipe out Lannister armies pretty easy after the Dothraki and her dragon made short work of the army they just encountered. If Jon brings proof, let him show Cersei and Dany shouldn't have to worry about being attacked but I doubt she was concerned with that anyways.
