I think it does though, because there is a difference between Winter and the WW's. There have been winters in the past that, ones that Ned lived through but it has been thousands of years since a WW invasion. Winter may not come to Essos but the WW army probably does. Also, the reason no one mentions it is because no one hardly even mentions it in Westeros, because it's been so long that it's just a folk tale to them. In a land where they don't even see winter, why would they even consider the thought of the snow zombies, lol.
Guaranteed it affects Essos and imagine how shitty of an ending that would be to the viewer, oh, lets just sail there, all us important people are safe now, thanks for the ships Euron, thanks iron fleet, tough luck Night King, maybe in a thousand more.
I'd be alright with an ending like that as well.