A child who receives US citizenship at birth CANNOT LOSE HIS CITIZENSHIP BY THE ACTIONS OF HIS PARENTS, OR ONE PARENT.
You need not blast away with ALL CAPS. Of course, you are correct. But, Obama would have been required to assert his birthright, and apparently he did not do so, at least not right away. He took a pause to turn the situation into an advantage for himself. Later, when he decided to turn to politics, he had to do some tidying up. It wouldn’t do for a Presidential candidate to turn out to have been a foreign student in his undergraduate days. Fortunately for him, he had powerful and resourceful friends and mentors.