No, it's part of human nature to be concerned about threats to ourselves above almost everything else. It's not only natural, but it's probably not possible to fully turn off this tendency.

Claiming that people should automatically be as concerned about being a danger to others as they are about danger to themselves is a nice thought... the world would be a good place if that was the way things worked... but it's not. Even the most educated and enlightened people put their own safety first and the threat to others second, because we're all human.