Mallory, Have you checked to see if your child qualifies for the state health insurance program? This is different than Medicaid and is provided to you for little or no cost. Every state has insurance like this for minor children. You may never use it accept for emergencies or a serious illness but at least your child would be covered.
I also have a wild 2 year old. I am surprised he has not broken any bones or had stitches yet. Just in case, I keep superglue and splints on hand.
As for my intuition, its very useful when it comes to my children, mostly I pick up on illness, especially vomiting. So far I havent had any feelings of broken bones or flying of the slide in the backyard.