Moat births are not as memorable as those which might cause a recall of the facts attendant to the birth of a future POTUS. I am a retired cop. I can make reference to several cases that were routine or not particularly significant upon my initial involvement. Later that individual or some circumstance related to the original case would assume a high profile and cause me to search my memory bank and recall the original players.
I find that no one yet has made such a similar connection to Hussein’s alleged birth in the US exetremely improbable. Not impossible but improbable, as is the likelihood that Hussein is a constitutional Natural Born Citizen.
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.