Regarding the law and Neglegence:
I do not know about Ohio or other states but in Fl the law states that any person under the age of 16 operating an ATV shall have a DOT approved helmet and eye protection. That's it. Nothing about size of engine or weight.
I think you would be neglegent to let you child operate an ATV without supervision. I do not think that the state would press to have your kids taken away for neglegence in the event you were present, the child was wearing approved gear, you were not on a road or highway, and the child crashed at a low rate of speed and was injured.
Based on the very limited information about the OHIO case, I believe the state was right.

Just my opinion,
JustMe