Even if birth records had been kept before 1900, because of the above reasons, there probably wouldn't have been a report. My grandparents obtained their birth certificates in the 1930s, through special application . These very important records are seldom used. Both civil and criminal court files are indexed. The United States court system is a bit complex for discussion here but your public library will have helpful books.
Files contain information on the children, their ages and personal data on the divorcing couple.
Having had occassion to bank finance a number of divorces over the years, I can safely say some of these files could provide an entire chapter for your family history book, if there was such an event and you were so inclined. Many court clerks in recent years have established a separate index for divorce records but in earlier years they can be found in the civil court files. These will sometimes give relationships. Individual veterans are responsibile for filing their records, but their unit administration forwarded the individuals records to the personnel center. I checked my records once and found they weren't accurate.