Definitely an interesting point. Is there any information on how much the risk of a crash increases if you wear a helmet?
I'm not sure that I agree it's in your child's best interest to increase the risk of you getting badly injured if you crash or fall by not wearing a helmet, even if this means that your child is less likely to be in a crash at all. I think it is important for my child to have a healthy mother. It's going to be scant consolation to them if I end up with a brain injury or dead because I was not wearing a helmet if things do go wrong, surely?!
Also, the combined likelihood of injury for you both must be lower if you're both wearing helmets, even if it slightly increases the risk of something happening in the first place? As in: both wear helmets, 1 % chance of permanent brain injury for both in a crash, 1% chance of a crash. Only child wears a helmet, child has 1% chance of brain injury in a crash, parent has 10% chance of brain injury in a crash, 0.8% chance of a crash.
In the first scenario, both parent and child have a 0.01% chance of brain injury. In the second, the child has a 0.008% risk of brain injury, but the parent's risk is 0.08%. I've made these figures up, but intuitively I'd say that the increase in risk to the parent from not wearing a helmet is much bigger than the reduction in risk to the child from the reduced risk of a crash due to the parent not wearing a helmet. I suppose your choice depends on whether you value your child's safety alone, or your own as well. As I've argued, I think my own safety matters for my child too.
Also, what are you basing the assumption that your child will be injured or possibly killed in a crash/ fall, even with a helmet on, on, whilst you only risk a head injury, whilst not wearing a helmet? Surely the comparison is.
Lastly, has any research been done on drivers' perceptions of people cycling with children? I imagine that might influence their attitudes and therefore the risks to cyclists as well.