Yeah I'm probably thinking way to much about it. I can't see the night king flying to Kings landing on his own as he generally doesn't do a lot of killing does he? That's what the whites are for and there's nothing to reincarnate if he killed them with dragon fire.
It would make a good story though, Him attacking Kings Landing the same time as the great war would be fun.
He's not in a rush either. Got nothing but time!
The general tactics they've had to date are using there numbers to overwhelm the opposition, not caring if they lose numbers due to their seemingly infinite supply.
There is a rumour about him heading for the place he was created on the island with the tree inhabited by the children of the forest, so there could be that.
The aim of those at Winterfell seems to be to kill him and hope the rest drop dead, so if he's not there it would be the best tactically, they just don't seem to be tactical in their approach. He is hell bent on destruction slowly walking across the world bringing it about.