I've not once feared for my own safety. Children as young as ~5 or 6 walk on my street alone or in small groups regularly, and they, likewise, have no reason to fear. There's even a culture of teaching your kid at a very young age to do things on their own. A regular TV show documents it (link with English sub: https://www.dailymotion.com/video/x4lme5a).
I spent 3 and a half years in San Francisco, and the difference is stark. People riding the bus grab their phones hard with one hand while using it with the other, for fear of it being stolen (as happened to a friend of mine). You would never leave your phone on a table while going to the bathroom unless you were with someone to watch it. If you lost your wallet, you'd be lucky if you got your driver's license back in the mail.
Are American parents overbearing as a group? Likely. But it's easy to criticize from afar if you've never lived in a low-trust society.