Something else had to have happened.
I'd hazard that on the whole Americans are less tolerant of risk than they were 60 years ago.
As an explanation, I'd tenuously tie it to a decrease in "real world physics" employment (e.g. farming, manufacturing). When 50% of the population works in farming (circa 1870), they have one way of evaluating risk and human capabilities. When that number drops to 2%, it's easy to forget how robust humans are.
Obviously, we're increasingly aware of the risks (usually without any appreciation of the odds). Previously, you heard through close circles. Now, by the time you're a parent, you've heard of hundreds of ways your child might die through news, movies, etc.
I'm a parent of three, and I actively work to minimise risk amongst my kids where I think it's appropriate, but I'm always conscious of when I'm doing it and try to balance it.
If news is optimized for clicks / views, negative stories are more engaging, and the availability heuristic is accurate (aka we assign greater importance to things we can immediately recall), then viewing profit-on-engagement news fundamentally reshapes our opinion of the world.
Historically, someone might've had 5-10 children and it acted as a form of redundancy. I think that feeling has changed significantly!
Conversely, I think there is a greater emphasis on extra-curricular activities now. These days, there is a feeling that you're selling your kid short or failing to set them up well if you don't have them enrolled in swimming lessons from maybe two years of age, in dance or sports or kinder-gym classes around 3-5 years of age, and then in after-school sports programs once they start school.