Teach your kids to not challenge Master Blaster to to-the-death combat in Thunderdome if it really worries you. My concern has never, ever been that they'll get run over. Not by age 7 or 8, when I would consider letting them wander unsupervised. There are many traffic fatalities in the US each year, but those occur not to people crossing suburban streets (or rural roads). Hell, I don't think they much happen when people are darting across downtown streets either.
It is a combination of abduction anxiety plus people who are worried the neighbors will call CPS on them. Those latter sort might be able to break the cultural norm that's forming, but there have been high-profile cases of people siccing the authorities on them for letting their kids be independent. I think any survey not specifically designed to lead them to other conclusions would discover this to be the case.
Or is this more r/fuckcars propaganda, and I've lost the plot? Are we all nostalgic for the days when only party officials could ever afford a Lada or Trebant, and the proletariat had more important things to worry about?
They are not key determinants. There are places in america you will see kids outside doing things. New urbanist developments, core of college towns, neighborhoods in big cities, ... And this is because stuff worth doing for kids is safely accessible without car there.
The big intellectual tragedy of the 2nd half of the 20th century is that we have become blind to the fact that much of our behavior, aspirations, dispositions are technologically determined, and not the other way around.