I hardly think that was the case. The information is kept on a form (usually on one of those hospital clipboards where the nurses are jotting down the patient's medical records). Then after the birth, the nurse asks the parents to help her with information to fill in the blanks with baby's name, etc. No one carried a typewriter around from hospital room to room filling in blanks. The form, filled out by hand, was then given to a clerk (typist) to type up. The typed form was presented for signatures, then filed with the state.
That’s so 1930s; by the 1960s they had typewriters. The only thing produced in handwriting was the signatures and dates (attesting to the signature). By the late 1970s they were back to a note sheet with keyboard entry into a data base system or word processor. By the mid 1980s word processors had swept all before them.
You're right.
Back in the Bronze Age, I worked as a secretary and typed up countless forms under exactly the scenario you present.
Rarely, if ever, was anything submitted for typing until all the information was available. What would be the point?
It's not only inefficient, it increases the probability of documents being lost. Picture this: a partially completed handwritten form is submitted to the typist. The partially completed typed form is then set aside and gets buried in a pile of similarly partially typed forms.
Now what? Does the typist go through the growing pile of forms each day to determine what's missing and then try to chase down that information? If this were standard operating procedure, it would be chaotic. The more the form is handled, the likelier the possibility of its being misplaced or lost.
It's an absurd scenario.