Interesting thread. To the OP, do you want to meet a neighbor and possibly be friends with him or just get involved in what you feel is an unsafe situation.
I actually read this forum to educate myself. No two situations are exact, but I want to know what went wrong. I have many times held a child while mowing on a garden tractor. I have keep their legs between mine and had an arm around them. I have only flat areas of moving and very few trees. Like others have said, I feel they are safer in this manner than playing in the yard while I mow. I am able to teach them safety. They know what they can and cannot do. If I'm mowing, they know they cannot be anywhere close. They know if they need to get my attention, they don't come towards me.
I have never seen any safety accident article that indicates this is a dangerous activity. I have read MANY articles on kids not knowing mower safety and they get run over, or slip under the blades. I do not let my kids sit in the bucket of the large tractor or ride that if I have an implement on. I have read many accidents which relate to those activities. The person bringing up sports is correct. Too many people feel it's acceptable for the kids to have life changing accidents in sports, but not in other things. I feel that accidents should be minimized to the best of our ability, but risks are a part of life. Kids should be learning.