Parents can and do change their childs citizenship all the time.
Mebbe so. But, so what?
The parents cannot renounce their minor child's birthright to US citizenship.
As your very own assertion confirms.
And Citizen AT birth is not the only require of being a natural born Citizen.
A child whose parents did give up their citizenship do have a right to ‘reclaim’ their citizenship. But they must actually reclaim it. And the process is one of naturalization. Thus, they are naturalized and no longer ‘natural born’.
This is Obama’s likely problem. If he did properly reclaim his citizenship he citizenship is via naturalization.
Be that as it may, we have the State Department documents indicating l'il Soebarkah was no longer American. This process usually involves an interview of the child by a State Department official. So, it may have been wee Soebarkah himself who denounced his citizenship.