I think this family story is plausible. She was born a US citizen and would have been aware that the 1920 election was the first in which women would be allowed to vote in a Federal election. If she tried to register to vote she would have been rejected as an alien and ineligible to vote.
I would probably be searching newspapers which should have carried stories of women in a similar situation. The more evidence you have that women discovering their citizenship status was "a thing" ca. 1920, the more plausible the story becomes.
