From my experience, two reasons.
1. Most docs and nurses don't remember the names or details of deliveries (except for medical details) unless they have an ongoing relationship with the mother or child. I only delivered 20 or 30 babies, and I wouldn't know if I delivered someone famous or not. I can only imagine the difficult remembering for an OB GYN who has done thousands of deliveries.
2. It is illegal to give out any medical information - including identifying someone as a patient - without a release from the patient. Even death doesn't change that, you need a release from the administrator or executor of the estate.
In 1961 the birth of a "black" baby to a white mother named Stanley would be sufficiently unusual to stick in most people's minds. They might have brought that memory out to "look at it", until something, say the canidacy of BHO. a "black" man with a white mother born in Hawaii, would trigger the memory.