I’m not familiar with all the SS laws/regulations, but I remember that the time/method of issuance changed at some time during the period 1950-1990. Before the change, you only applied for a SS# when you needed one for your first job. I got mine around age 13-14.
I don’t know for sure, but I think that after the SS law/regulations changed, weren’t SS#s issued at birth and would therefore reveal the location of birth?
Being born outside of the US and living for years outside the US would account for a late application for a SS#, but wouldn’t necessarily exclude a person from the Presidency if the parent(s) were US citizens, or would it?
I got mine in Illinois at 13, in 1963. I still have it in my wallet, in a laminated state of preservation.
My memory about that is fuzzy, but that's about how I remember it. I think I got mine about the same age so I could have a paper route.
I was adopted shortly after and I remember having to go back to the court house for something to do with SS, I think it was to get a new SSN, IIRC.