I now live in the US and have kids about the same age.
The issue is not homogeneous culture or any other thing people with no brains come up with, but traffic density, at least in my case.
Where I used to live we could roam without having to cross dangerous streets. Where I live now, there's traffic everywhere and cars don't give a shit about people on foot, so my kids do not roam.
When I was a kid, we used to have vast areas to play safely. I just looked on Google Maps and the cross-country ski route we used to regularly take on weekends is 4 miles long. We had to cross ONE street. I used to bike about 10 miles one way, by myself, when I was a little older and keep on bike lanes the entire trip.
We also used to have large play areas in the backyard of every home I ever had. The bigger ones were the size of a typical New York borough block. We didn't have to leave our homes to get to a playground, or a park...we had one right outside of our door.
Obviously this sort of thing is easier to do / plan for when the entire country has less people than the city of New York.
(weird really - can you do that for killing people by any other means?)
My wife and I have semi-seriously considered moving to northern Scandinavia for both this reason and its comparative likelihood of remaining a decent (well, bearable, if only just) place to live in a clathrate-gun scenario.
There was a case not far from where I live where a driver turning right mowed down a 4-year-old child walking with his grandmother on a CROSSWALK. He wasn't running or anything like that, just crossing the street. The child died. They didn't even charge the driver. Fucking ridiculous. It was 100% the driver's fault. New York Times published an article written by the child's mother who questioned the logic of first of all call it an accident and second of all how the driver wasn't charged with anything, not even reckless driving.