My experience is that almost any government handout becomes a scam target with the pros figuring out all the angles to get their best return for their "effort". In this case American citizenship was a brass ring worth scamming for regardless of mom's hopes for her newborn. IMHO. ;-)
Well supposedly you couldn't just phone in the notice. They were triggered by filing of the certificates of live birth. But those could be filed without a doctor's signature, just need a "witness". Mom might not have actually signed, if she was out of the country. But even in 1961 you could call from Vancouver, and with more difficulty from Mombassa, with the information. Grandma would need the sex and a name and she could forge the parents signature. Although she could have signed as "informant" and had someone else sign as "witness". That way there would be an official long form, which might or might not show the actual birth location. The newspaper announcement contains no location, and in fact not even a name for the child, just "son".