The problem was not proof of age when applying for SS; it was proving one was old enough for benefits when applying to receive them. I filled out many such certificates 30 years ago. Issuing SS at birth removes that need.
One of the WPA projects was to index past US census returns to help individuals prove their age. As census takers (and individuals) were not always the most accurate when recording names, a system to allow for sound-alike surnames to be grouped together - the Soundex - was developed and used to index portions of the 1880, 1900, 1910, 1920, and 1930 census.
History of the Soundex (census.gov)
Soundex - FamilySearch.org