She could have, but it would have been totally unnecessary under immigration law at the time. Since his mother was a US citizen, her grandson was eligible to receive US citizenship any time before his 18th birthday.
Now why would grandma commit purjury and risk jail time to obtain something that could be easily obtained legally?
You assume grandma might have had sufficient legal knowledge of immigration law to make it seem unnecessary to her to make any false report of a foreign birth as an HI home birth. People panic!
There was one report (from my questionable memory) I think from a friend of Stanley Ann's (before the friend clammed up) that Madelyn was scared to death of the Mau Mau uprising in Kenya and wanted to protect little Barry.
If there were to be custody issues resulting from a fall-out between Stanley Ann and BHO Sr, especially if there was a Kenya birth, BHO Sr might have had a stronger claim for Kenya jurisdiction of custody if Barry was not a US citizen. We know there was a big fall-out in the marriage because one month after the birth SADO was living in Seattle! Madelyn might have filed US birth papers for Barry and sent SADO to Seattle for protection from BHO Sr who was reportedly a beast.
The point might have been not to just confirm citizenship for Barry after a Kenya birth, but to actually _negate_ the Kenya birth to disarm Kenya claims on Barry.