There's reason to believe that contemporary American child-rearing gets some things better than other countries. There's also reason to believe it gets other things worse. On the one hand, American gender roles are relatively egalitarian. However limiting and intellectually repressive parenthood may be in America, it's much more restrictive for mothers in traditionalist gender-segregated societies like Japan, Italy and Greece. That comes out in childbirth statistics: women in Japan, Italy and Greece have simply stopped having children. In other societies with gender-segregated traditional family roles, like Vietnam, higher birthrates result from intense Confucian pro-natalist social pressures that leave women extremely unhappy, and birthrates there are likely to drop rapidly as women achieve greater social independence. American women, meanwhile, are still choosing to have kids, and that's partly because they can continue to have careers, and their male partners share at least some of the child-rearing duties.
On the other hand, as Ms Senior writes, America's lack of paid parental leave or subsidised day care makes parenthood much more stressful than in similarly wealthy France or the Scandinavian countries. In part, the anxiety and over-protectiveness of American parents criticised in Lenore Skenazy's FreeRangeKids blog stems simply from the absence of such support systems. But it's always seemed to me that this anxiety is also driven in part by high levels of inequality. In a society with a large gap between excellent and inadequate schools, parents face tremendous psychological pressure to raise and educate their kids the "right" way. In societies with a more egalitarian distribution, parents don't reproach themselves so much for laying off the kids a bit.
This, I believe, also explains why in highly egalitarian Australia, child-rearing consists of turning the tykes loose barefoot in the backyard for 12 years and hurling them slabs of meat thrice daily. They seem to turn out pretty well, actually.