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.
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.
Abductors don't need to be everywhere. They just have to be a plausible threat to affect the thought process of a parent that's deciding whether or not they are going to leave their child completely unsupervised and unattended.
Maybe if the media didn't have a fetish for covering the incidents that do happen. Maybe if there wasn't a vocal subculture that crucifies parents for leaving their children to their own devices. Maybe if the thought of having the center of your world being beaten, violated and murdered at the hands of a stranger wasn't so mind-numbingly horrific. Maybe then this trend would start going the other way.
Or not? Sure parents used to let their kids roam free. They also let them ride in the back of pickup trucks and beat their asses when they did something wrong. There's probably a healthy middle ground.
What I would suggest then is that, those that are left to roam are in more danger than those who aren't, because there aren't as many alternate targets. There are also fewer witnesses now too. So even if you think it is a big scare, there is less, to use a vaccination analogue (incorrectly I know), herd immunity.