It seems like the DOH rules allowed for a very smooth rhythm. I would imagine that if there were several clerks in the registrar’s office, there would have been one who did birth certs, one who did death certs, and another who did marriages. The person would know what they were looking for, the regs they had to follow, etc.
I could see that person sorting out all the certs received that day that were complete and stamping them with certificate numbers, bing, bing, bing and then date, bing bing, bing. Then looking at the incomplete certs or certs that had affidavits attached and filing them however they had to be done, whether requesting more information or whatever. If everything checks out within that day, stamp it with the next number and the date.