In 2023 my wife and our children became German citizens via a declaration process called Staatsangehörigkeit § 5. The official paperwork came in the form of a document, an Urkunde über den Erwerb der deutschen Staatsangehörigkeit durch Erklärung, which we used to apply for Reisepässe.
The Urkunde durch Erklärung is very important, and will be needed to renew the passports. It is possible to replace it if something happens like fire or theft, but it isn't very straightforward to obtain the replacement. We decided to additionally file for birth certificates in Germany. These would be straightforward to re-order in the future as needed, and would serve as proof of citizenship. While at it, we also registered our marriage.
A civil records office in Germany is called a Standesamt, and registrations of foreign births are handled by Standesamt 1 in Berlin. The Berlin Standesamt 1 is famously backed up in processing submissions, we were advised to expect 2-3 years to process our forms.
Happily though, it effectively only took 5 months.

I say "effectively" because we missed the email of invoices to pay the Standesamt, until the Consulate sent them again 3.5 months later. So overall it took 8.5 months, three and a half months of which was on us.
Jun 17, 2024 | Submitted forms at San Francisco Consulate. |
Aug 2, 2024 | Consulate forwards invoices from Standesamt in Berlin, which we missed seeing. |
Nov 18, 2024 | Consulate re-sends the invoices from Standesamt in Berlin, we paid the next day. |
Jan 17, 2025 | Recording date listed on the certificates. |
Feb 7, 2025 | Consulate receives the certificates from Germany. |
Feb 28, 2025 | Certificates delivered to us. |
If you decide to do something similar, be aware that it is an expensive undertaking. Registering four births and one marriage cost 334 US Dollars in fees to the Consulate and a total of 630 Euros to Berlin Standesamt I. Altogether, the fees came to about a thousand US Dollars. One is not required to register the births in Germany, only do so if you believe it will be worth it.