As a frequent visitor to Europe, one of the things that appeals to many/most people is the history, art & sophistication. Euros in general seem to be more polished, less rough around the edges; especially around their uncouth cousins, the Americans. And yet, like clockwork, the line up and kill each other by the millions.
I've actually come to the conclusion Euros are incapable of self-government. The average commoner has been manipulated and controlled for so long, that they don't have any instinct for self preservation. So, they enjoy life in all its trappings until their "boss" tells them it's time to party.
The US is definitely rougher and more independent. Like you mention, citizens have skirmishes with loaded weapons, resulting in higher homicide stats. But, unlike the Euros, at the end of the day, everyone in the US - 330m - speak the same language and abide by the same cultural standards.
It sort of like bend, but don't break. The Euros are beautiful, but tend to break.
Europeans have a greater sense of culture, language, and borders than Americans. Why do you think there are so many countries in Europe despite hundreds of years of wars and conquest? Underneath the thin veneer of the EU, Europeans have a strong sense of national identity and survival. The rise of nationalism in Europe today reflects the growing opposition to globalism and the loss of sovereignty. Europe is breaking up into smaller and smaller nation states and has been for years.