You really, REALLY wouldn't want to be the first person to kill a zombie when the outbreak starts. Everybody's going to think you just shot someone in the head. Actually, that's exactly what happens in the first season of Fear the Walking Dead. A zombie bites a cop, and the other cops kill it. But every witness thinks the cops just killed a homeless guy. That's when the riot starts.
Even if the outbreak is contained, people would take advantage of it by just killing anyone they don't like. All they'd have to say is "he turned and I had to kill him."