I think we can both agree the AAP language is a bit wishy washy and doesn't clearly push one view over another. And for me that's not good enough. Before I take a knife to my child I'd want the medical authorities to scream loudly "you must do this or your child will be harmed!".
The AAP aren't necessarily suggesting the Government pay for it. Infact there is a big movement in the US against medicaid covering circumcision as it is an unecessary procedure.
By third party I am sure the AAP are referring to Private health insurance ... Which works quite differently in the US. Most things are covered are no/very little cost to people... If they can afford health insurance. There isn't a lot that health insurance doesn't cover. So saying a third party should cover it doesn't support your cause. It just reiterates that the US pirate health insurance system is better than ours in that it covers nearly everything.