Yeah, considering the US have a bunch of problems related to violence (mass shootings, overzealous police force, invading other countries left and right), it makes sense that other violent processes are also problematic in the same environment. I agree with you death penalty is merely a symptom of something else in the whole system of the country.
Makes it extra weird when you consider how prude the US is when it comes to sexuality, while violence is something that is not only used as entertainment, but even many feel pride about.