not a citizen at the time of his birth?
Here is my answer:
Well, the concern rests with the wording in the constitution for the requirement of President.
The writers (George Washington being one) of the constitution were concerned that someone would come to the country, gain citizenship or be born to a non-citizen and have dual citizenship (a person born of parent who are from different nations is a usually a dual citizen) and thereby the person might not be loyal to America-alone. So an extra step to safeguard our nation from a takeover was made: to require a person be a NATURAL BORN CITIZEN in order to become President. (you have to be careful with the definition of NBC because many people have tried to change its meaning to weaken our Nation's Sovereignty and to exploit America to gain financial benefits.)
The constitution requires for the president to be a Natural Born Citizen, which means both parents must be citizens at the time of the birth of the child and that the child was not born outside of America and thereby obtaining citizenship of another nation.
No one in our history has tried to become president that was not a Natural Born Citizen, so people have not really focused on this requirement since no person has tested its limit and pushed its meaning to the edge. The discussion gets muddy in the fact that there are a court few cases that Judges have carelessly used the term Natural Born Citizen in the writing of their judgement dealing with citizenship rights, like the right to vote for women. You do not have to be a natural born citizen to vote. You don't need to be a natural born citizen for anything else in America! There is only one legal Constitution time in all of American History of when and why you need to be specifically a natural born citizen and that is the higher standard used as a requirement for a person to become president.
Since the Justices have carelessly declared people to be a Natural Born Citizen for a benefit or right in legal writings many people try use these cases as a legal precedent to define what a Natural Born Citizen is also for the requirements of becoming president, but it must be carefully noted that no justice has ever tried a case on whether a person is legally eligible under the Natural Born Citizenship requirement written in the Constitution regarding a president. This is a very specific measure and it is only meant for this single purpose. Simply because judges have used the wording inappropriately to apply to a judgement in a case to make it sound good does not change the Constitution meaning of the requirement for a person to become President or the intent for the precise wording used.
When America began, everyone here was not Natural born citizens - obviously - because we were not yet a nation able to give citizenship. People were citizens of Britain since we were a colony, after the revolutionary battle against Britain for so many years a loyalty was forged among most who were still in the country so anyone living in America at the time of the constitution's ratification were the original citizens and thereby eligible citizen for President (similar to what we legal call a grandfathering clause) but anyone after the writing of the Constitution and its ratification needed to be born of two American Citizens and to be born in America (military bases are considered American territory for this purpose).
There has been a great fight to lower the standard. For example, since we have so many people who get to claim to be citizens as anchor babies it is obvious that we would not want a child born of two mexican parents whose mother happen to have a baby while she was legally or illegally in America to be eligible to be president because there would be a danger that the child's loyalty would be with Mexico or at least influenced in a bias toward that other country.
Another case is that many Arabs citizens comes to school in America for college. Countless of them marry an America girl and have a child simply to become an "American citizen" and to have a child with "American citizenship." I put it in quotes because of course you could see the problem with giving such a person the right to call themselves a natural born citizen. Most divorce and abandon with women and some even take the child after a year or so, but dangerously they and the child are citizens but thankful neither or Natural Born Citizens!
This is similar to what happen with Obama. His father was a citizen of Kenya which was a colony of Britain at that time, so his father was had British citizenship through Kenya (which later was released and able to give their own Kenyan citizenship so he was later a citizen of Kenya). His mother was under the legal age to have sex with an adult - legally she was a victim of 'statutory rape' because she was not married and she pregnant with an adult's child while she was a minor. (she later married him; we are told before the birth of the child). Obama Sr was never charged with statutory rape, but he could have been legally.
Obama's mother was not a very patriotic person, she was in fact anti-American and supported Communism during the height of Communism in the World--not that this effects HER legal citizenship, but it adds deeper concern to Obama's heritage and allegiance and citizenship. Of course the big question is made out to be where was Obama born. Frankly, it does not really matter because he is not a natural born citizen anyway - his father was British/Kenyan... (other interesting but not legal side notes: she lived in HI that had just become a State ... many people in HI did wanted to be a state of America but due to WWII there was such a military presence and it just seemed obvious to make the islands a State since it was a critical place to be attacked by larger aggressive countries.)
But another legal citizenship concern is: His mom also left America and married an Indonesia man. By Indonesia law she would have to renounce her American citizenship to become an Indonesian wife and her children would also become Indonesian, and loose his citizenship. So again Obama, Jr's citizenship is in question because there is no record of him apply for citizenship after he returned to America years later.
I know I answered much more than you asked but I hope this helps and give you information to maybe do your own further investigation.