I live in a suburban R-1 neighborhood in Oregon and my kids play basketball, ride bikes, and roller skate in the street literally every day with boatloads of other kids. If that's your vision of a nightmare, I fear we're not going to agree on any of this.
I'm seriously shocked by how many people value the idea that kids can play in the street. Why not a park or other recreation area? I get that it isn't entirely common in the United States to build walkable parks but that is a United States problem, not a city problem. Cities built for living should offer much better alternatives to children than playing in the street.
We have a great park just down the street (about a block and a half), but the street is great and it's right outside. Maybe 10 cars per day travel our street. Besides, the street is paved for bikes and rollerskates. The park is, of course, not. Also, it's easier to keep tabs on the kids right out the front door than it is when they're unsupervised down at the park.
But, yes, they also play at the park. They all run all over the damned place -- because it's a safe place and, culturally, the people living in the neighborhood have a high degree of trust and look out for one another. It's very difficult to replicate that in high density urban developments with the attendant problems those areas recruit (e.g., crime, homelessness, etc.).
How do they get there? What might not be obvious is that the parks are often far away, because you can't afford to maintain many parks because the tax base is too small.
I completely agree if we are talking about a typical suburb. A mixed-use area built for living doesn't have that issue. What I find really strange is that some people in these comments seem opposed to mixed-use areas BECAUSE their children will no longer be able to play in the streets. It's like some sort of weird, cyclical way of justifying the lack of proper pedestrian areas.
If the tax base is too small for parks, it's also too small to pay for the street/road infrastructure, and that's being subsidized by people outside of the community. If a community can't afford basic infrastructure (which should include parks), it isn't zoned to be dense enough.