There’s another reason hospitals would not pre-assign numbers for birth certificates ... you might not have a live birth. In such a case, the baby would not receive a certificate of live birth, but a death certificate. There not going to assign numbers until they know the outcome of the birth.
Excellent point.
As one who was admitted to the hospital so we could induce labor for our overdue daughter, only to have her die during monitoring, I am keenly aware that not all births are live births. That was 17 years ago but sometimes it feels like yesterday.
As I was looking at newspaper announcements from 1961 I was struck by the number of children who died, including infants. A lot of pain represented in each 2-line announcement.