The official SA Govt records were compiled from local church records, despatched on an adhoc basis to Adelaide.
Your statement is not quite accurate. I usually link this interesting article but find that many don't bother to read links. There is also a book about the SA BDM's and its inherent problems in South Australia but unfortunately long out of print. I hope you find the following interesting and informative.
In July 1842 the colony was effectively divided into two District Registries based on Port Lincoln and Adelaide headed by
District Registrars answerable to the
Principal Registrar in Adelaide.
Every 6 months the District Registrar had to file a copy of his certificates with the Principal Registrar. All people living outside the defined areas had to make their way to the nearest Deputy Registrar.
Parents were compelled to register
births within 6 weeks or face a £10 fine. Some late registrations had their birth dates adjusted to comply with this requirement and avoid the fine, while other just failed to register. Some groups in the community and especially Catholics and Scots were reticent to register births.
The responsibility for reporting marriages lay with the celebrants who had to supply a return to the District Registrar every quarter.At the peak of the system there were 22 districts. Within each district,
agents were set up to service local areas. They were local people of some repute, such as a postal officer, a police officer or a landowner. Their task was to pass on notification to the
District Registrar who in turn created two certificates, one of which was retained and the other forwarded to the Principal Registrar as the official record. There are a considerable number of known differences between the two certificates.
There were many opportunities for events to be not registered or the registration to be lost in the chain of reporting.
Cheers

Cando