It would not take a "vast conspiracy", just his mother or grandmother filing an unattended birth certificate. That would generate the announcement in the papers(plural) and produce the exact same "Certification" that was posted. The reason they would do that would be to make him a citizen, not so he could become President 47 years later. They'd need to do that because at the time of his birth, his mother was not eligible to pass on citizenship, if he was born outside the country to a foreign national father, not having lived in the US for a sufficient number of years (5) after her 14th birthday, since she was not yet 19.
It appears that unknowingly, Obama, Sr. did us a big favor by passing on British Citizenship at birth to Jr., making it impossible for him to be a NBC.