I suspect this is not just an American or British or solely cultural thing but also a number-of-children thing. This seems like it would also be the natural response to having fewer children. If you have 3-10 kids, you can "afford" to let them off the leash so to speak, if only psychologically. Your kids going missing or getting hurt is an eventuality anyway, maybe. If you have 1 or 2 kids, you are investing a lot mo…
K-selection vs. N-selection is what you're talking about in evolutionary biology. Few offspring, high resources invested vs many offspring, lower resources invested per child.
https://en.m.wikipedia.org/wiki/R/K_selection_theory
(K/N selection doesn't perfectly describe modern kids, and it's not an ironclad model anyway, but evolutionary biologists have thought about this a decent amount.)