But where does it (reliably) say that the Others are “omnicidal”. To the COTF, the First Men must have seemed omnicidal, the two tried to exterminate each other. The Human-COTF war started out as a side effect of migration and cultural differences, they didn’t start killing each other just because, but it snowballed into attempts at genocide and human sacrifice. TWOTFK saw numerous atrocities on all sides, and even now “good factions” aren’t big on morality (Frey Pies), but is that omnicide?

From the pitch letter: 

“The greatest danger of all, however, comes from the north, from the icy wastes beyond the Wall, where half-forgotten demons out of legend, the inhuman Others, raise cold legions of the undead and neverborn and prepare to ride down on the winds of winter to extinguish everything that we would call “life.” The only thing that stands between the Seven Kingdoms and an endless night is the Wall…”

That’s Word of Author right there. The Others want to kill everything. Full-stop. And as @poorquentyn says, remember the bear. 

You can say a lot of things about the various participants in the War of Five Kings, but wanting to kill everything with warm blood isn’t one of them.