Actually, it's more complicated than your effort to sweep it all away would acknowledge.
IF (and it is true) the schools in Indonesia noted (on the school records) that young Barack's citizenship was "Indonesian,"
then it can only be true that EITHER (1) President Obama is Constitutionally ineligible to be President of the United States OR (2) the Indonesian school records were falsified by the Obama family AT the time of his application to be admitted into that school.
As to the former (number 1), IF the President at any time in his life was a citizen of Indonesia, which does not recognize dual citizenship, then in order for him to BECOME a U.S. citizen, he would have to be naturalized. (If he was a citizen of Indonesia who never got naturalized, that's even worse, for in that case he's STILL a citizen of Indonesia -- not a citizen of the United States). If he is a U.S. citizen because he got naturalized, then he is not a natural born U.S. citizen. And he would then be Constitutionally ineligible to be President.
{NOTE HOWEVER -- there is a very legitimate possible rebuttal to the foregoing musings. It is doubtful that the young Barack could "renounce" his U.S. citizenship based just on the actions of his parents, without more. SO, maybe none of that stuff ever did terminate his U.S. citizenship -- assuming he was born in Hawaii.}
But, as to the latter (number 2), IF the President's family lied about his claimed Indonesian citizenship on his school application papers, then we know that his family was willing and able to lie about his citizenship. This is why the Certification of Live Birth -- based on a shitload of hearsay type information -- was always less than persuasive. It is not difficult to imagine the pregnant mother of the future President being unable to fly and having to give birth in Kenya, instead. And it does not require a massive conspiracy theory to figure that she was bright enough to realize that this scenario might cause her newborn baby some future difficulties as to citizenship. (No thought of his being a possible President required to be concerned about the legal ramifications of his citizenship).