Just to clear the question.
I never heard anything about any birth certificate from anyone after I enlisted in 1949 and got to Lackland AFB. I told the soldier (when I enlisted I didn't know a private from a general) that I was 19 since I had been told as a child that I was born in 1930, and the army must have taken that as enough. I was just a young working cowboy then and I don't remember anyone in or before 1949 that had ever even asked my how old I was. I have never tried to get a birth certificate since I have never needed one. So I don't know if there ever was one.
I'm not sure that the military is actually much concerned anyway. When I got back from Korea and out of Madigan Army Hospital in 1953, and sent to Fairchild AFB, my Squadron First Sargent, a veteran of WW-II, told us once that he had enlisted when he was only 16 and the Army never knew it or didn't care. He had been a M/Sgt in 1941, on Gen MacArthur's staff and was a prisoner of the Japanese when they took Corregidor.