I don't see where you're from, but based on your comments, I'll take a guess: Are people wigging out because of concerns over mountain lions?
If so, just tell 'em to relax. Keep your kids close and stay alert so you don't stumble right up on one. The frequency of human attacks is very low, but they get news... Remind your critics how many people are attacked by Moose every year. Or vending machines.
If they're freaking out over something OTHER than mountain lions, then I'm lost for an explanation (unless you're in Moose country, see above). Animals, Reptiles and (for the most part) insects aren't much of a danger in North America. We have few poisonous critters, and those that are tend to be the non-aggressive type. Biting flying insects are a problem at certain latitudes, but none of those are lethal, just ranging from annoying to painful. Human-on-human violence in such low-population areas is rare, even more so when said humans are in a group.
Sounds to me like you're doing right by your little one.
Now back on topic: Outdoor Injuries and Close calls
I got into a hornet's nest while collecting firewood once. Yuck.
I've got a bum ankle from a childhood injury so I have re-sprained it on a number of occasions, but only some of those occasions were on-trail, the others were going down stairs in suburbia.
Our little one isn't walking yet and gets carried everywhere, so she hasn't suffered any injuries yet.