Think back to your favorite zombie movies. Now name one in which the zombies are focused upon as active villains. The fact is that rarely are zombies ever proactive in their roles. They shamble about (or in some films sprint about), merely responding to the world and affecting those they encounter. They are unthinking, uncalculating, and genuinely unintelligent. In effect, zombies serve more as a natural disaster than true monsters or villains. While devastating, they are never consciously stalking their victims. Their malevolence is no greater than that of a tornado, earthquake, or hurricane. They are an event that causes massive destruction in their path, to be sure, but they never do so with a willful intent to harm. The zombies of a film like Night of the Living Dead are no more villainous than the tornado in Twister or the earthquakes in Earthquake.
Like in any good disaster movie, the true focus of good zombie fiction is the people themselves. Most of the stories follow the same formula: a group of different people must band together to survive and hold on to some semblance of civilization. The human characters both spur the action and hinder it, playing antagonists to themselves. Almost every death in a zombie story comes about from some human action, whether from carelessness, unforeseen circumstances, or genuinely evil intent.
For example, consider Night of the Living Dead (SPOILER if you've been living under a rock and never seen the film). Every death can be traced to human actions. Barbara's brother dies from his choice to ignore the gardener. The teenagers die from Tom's carelessness. Harry is shot by Ben. The mother dies by choosing not to kill her daughter's reanimated corpse. Barbara dies from her inability to cope with the situation. Finally, Ben is shot by the posse clearing the fields, possibly with racist intent. In no case do the zombies have any direct role in a character's demise. In fact, the human characters themselves often behave in a far more loathsome manner than the zombies. Even in the realm of videogames, zombies can be traced back to human meddling. Look at Resident Evil and Dead Rising. Both provide human causes for the outbreaks. In a way, zombies are a manifestation of the destructive force of human nature.
Once the initial titillation of the gore and blood die down, a great zombie movie lives or dies on the human factor. How do survivors deal with a world diametric to the one they once knew? In Dawn of the Dead we see humans seeking solace in consumerism. In Shaun of the Dead we find them seeking refuge in familiar places. In 28 Days Later we see the soldiers use military order and brute strength as a means of dealing with, and taking advantage of, the new situation. In the Walking Dead graphic novels, we see countless human interactions influenced by the desperation of a lossing of all that is familiar. It is these moments that ring true to the audience. Zombies are a mindless natural force ushering in apocalypse. Their allure comes more from their external effects on the world than from any discernable trait they posses. The human experience, however, is something tangible, relatable, and infinitely compelling.
Whiles the zombies are what give "zombie fiction" its name, they aren't what give it its magic. "Zombie fiction" draws its power from the living, not the undead. We may initially come for the scares of shambling corpses, but the true fans stay for something far more lasting.