I admit I'm not familiar with how much the officer involved would be exposed to the names, but I can tell you from experience that medical people often have minimal contact with the names. A non-complicated delivery in the 1980's (sorry, no experience in the 1960's) meant delivering the patient who the doc may or may not have met previously, then two daily post-delivery rounds, which may or may not be done by the same doc, and one 6-week follow up. The delivery room nurses would not be the same as the ward nurses.
I remember the names of very few patients. Most of those I recall I either saw repeatedly over a long period, or were the very few who had something medically memorable.
I could have delivered someone who is now a celebrity, and I would not know it. I think you'll find most doctors and nurses say the same.
There is a space on the actual birth certificate (as opposed to the forged Certification Of Live Birth that Hussein proffers) for the attending physician signature or license number and the address for the attending physician or hospital.
I would think that someone would have taken at least a perfunctory glance at the legal document that they were signing. A name like Barack Hussein Obama Jr. is so unusual , particularly linked to the first black president that again, I find it highly improbable that no Hawaiian physician, nurse, registrar, or clerk involved is alive or that they are all dead, and not one has stepped forward to claim their place in history.
There are too many improbababilities that I have to swallow, particularly involving a POTUS with a record of prevarication that I have not seen in any president, including Bill Clinton. I would note that the only living human beings who claim to be present at his birth live in Kenya and say that he was born there.