It’s not the zombie virus that kills you; bites are lethal for a separate reason, most likely a host of bacteria and whatnot. The bites cause sepsis or whatever, and when you die, the virus does its thing. In The Walking Dead, Jim takes days to die of infection-like symptoms, and in Night of the Living Dead, the girl dies after developing a fever and slipping into a coma. In another movie which I can’t recall perfectly, the bite was shown with radiating, discolored veins leading from the wound, implying that the locus was infected. Zombie viruses being in the brain, I think we can conclude that the bite itself is lethal separate and apart from the reanimation cause.
Of course, the bites also transfer the virus, but that’s void in universes where everyone is infected.