Fascinating discussion, YY and CT. Pity it is hidden in this thread..
Anyway, YY it is clear you have never been to the US Embassy in Tel Aviv applying for citizenship/birth certificates for your kids.
We had to supply dates of when we lived in the US, and brought along lots of proofs like college transcripts and voting registration cards, etc. There is also no guarantee that if you proved it for your first kid, you don’t have to prove it all over again for the next. They may not always ask for everything, but those were the official instructions at the time when it was relevant for us.