Easy: Because Dr. Sinclair did the paperwork for Dr. West and filled out what was then an unremarkable birth certificate form. Doctors doing charting for their collegues is nothing new under the sun.
It is routine for nurses or other doctors to forge the signature of the attending physician if the latter cannot be found when the paperwork is being processed, but they wouldn’t put their own name on the document if, in fact, they weren’t the attending.