You did say this:
From post 96. l. She likely would have been a head nurse or someone like that. It’s likely that she would have been the one to stamp the date filed because it seems to coinside with the date the doctor signed the certificate.
That doesn't make sense. A nurse is NOT going to stamp the the "Date Filed by the Registrar" OR the "Date Accepted by the Registrar," which is OBVIOUSLY the date when the birth certificate goes into the state vital records as shown to you above and when it gets the control number. Now, why would a nurse accept or file the birth certificate for the state? Answer: the nurse would not.
Because at least one head nurse/administrator is likely a deputy registrar authorized by the state, much as a notary or a title clerk. And that nurse probably wouldn't assign the certificate number. She'd just sign the form and datestamp it once she verified all of the proper information was filled out.
Even butterdezillion acknowledges that a hospital employee was likely authorized as a local registrar.