I'm not a parent so I can only speak from the way I was raised. I was brought up much like your kids, only I never did leadline.
By the time I was old enough to hold my own head up, I was riding in the saddle in front of my Dad whenever he would warm up his roping or show horses. My first memory is of one of those times. As I got a little older, he would ride behind me but give me the reins so I learned how to steer. Then, when I had the steering part down, he (or my mother or older brother, whoever I was riding with) would sit static and basically be a passenger while I was controlling the horse's speed and direction.
When I was 3, I got my own mount, which was a mini burro and we learned a lot from each other. Then, when I was 4, my brother got a new horse so I inherited his old one, a 16hh, 1500 pound roping horse named Buck. He was 100%, completely bombproof. I rode him for a lot of years and never got hurt on him.
IMHO, if you take the proper precautions and don't turn the kid loose on a horse before you believe they're ready, then to heck with what anyone else might think.