That is not civilized. That is not the lesson we want to teach. Just because people feel fear and hatred and the desire to kill does not mean they have to act on that desire to kill. You failed to address the biggest reason some people are against the death penalty, that there is a very real chance of the State executing innocent people. As it is now, the odds are stacked against most people accused of a capital crime.
People do make mistakes.
Most of the accused have little money for lawyers. Unpaid or very low-paid lawyers do not have the resources they need to properly defend their clients. In many cases, they also don't have the will to do an excellent job. The people in town, the lawyers' clients and potential clients, are horrified and scared from the violence of the crime. When they see a lawyer defending a man at whom the police and press have already pointed their fingers, they will turn against the lawyer, and the lawyer's business may dry up.