Does she sound to you like a modern woman?
How many would have opted for birth control were it more readily available and effective?
Consider that your projections don't apply to people from a 100 years ago.
If you went back in a time machine, and said “here’s some medicine, take it if you wanna stop at three kids” and changed nothing else societally, would everyone who had 16 kids still opt to have 16?
I think it’s unlikely every single woman would make the same choice given an alternative. Modern or otherwise.
That should answer the silly question.
She's not a character from a novel written by a modern day feminist about how hard and traumatic life is in a Middle Eastern country because she has to carry the burden of raising children.
The number of people who had 16 and didn’t want 16 is also non-zero.
Availability of birth control coinciding with a fall in number of children per family suggests what the balance between the two might look like.