Not really. Just look at all the different burial cultures throughout history. I could easily see a society developing that stabbed a body in the head after death as a normal part of morning. In fact it would make for an interesting story to visit the planet hundreds of years in the future. The zombies are all gone, humanity seems normal except for this one practice of stabbing dead bodies in the head. Then the time traveler/visitor comes and questions that. Next thing you know he's reintroduced zombies to the world.
Yes, and if someone died quietly of old age in their home, or in a massive car/plane wreck, or overdosed, or from a heart attack, or any other sort of death where someone else wasn't around to shoot them in the head, them blam, welcome to Zombie Apocalypse 2.0... then 3.0... then 4.0... etc.
It would be a constant, never-ending concern. Society would, in no way shape or form, function as it does now. There'd be no hope for "normality" if, whenever anyone died, they would reanimate as a brain-eating monster from Hell. Even if everyone carried a firearm with them 24/7, there'd still be a constant threat. The never-ending paranoia alone would all but insure that innocent people were constantly being killed by non-zombies.