No, they could not. At 6 or 8 or 10, you cannot sign contracts. You can not fill out paperwork knowingly rejecting US citizenship.
If you want to spend the rest of your life searching for paperwork filed with the State Dept, where Obama at age 7 rejected his US citizenship, good luck! But if you ever find any, no court will uphold it.
Do you have kids? Can a 10 year old sign a binding contract on his own behalf?
Never the less, first you said this:
“At a minimum, it would require Obama to have gone to an embassy in a foreign country, and then fill out paperwork and swear he was renouncing his country.”
So you have admitted that there are regulations that allow him as a kid to fill out paperwork to swear and renounce his U.S. citizenship.
You then say:
“no court will uphold it.”
Well if there are regulations in place that allow him to renounce his U.S. citizenship, even as a kid, then the court must uphold it.