For perspective, my grand grand mother had 16 kids. All her grand grand children who are my generation that I know of have either two or three, with no apparent plans to have any more.
And if you ask any mother with >4 children she will tell you that 3 children is the maximal burden in terms of work. Once you have more than 4 children the job gets easier, not harder. Experience, older children pitching in, being forced to drop what's not important, et cetera. One father described it as "zone defense" rather than "man-on-man defense".
But the dollar cost does not go down. Time off work + $250K per child mean that only the rich can have more than 2 children.
Even things like private schooling levels off (many have a “all kids after three or four are free). The biggest one that doesn’t level off would be college.
Many health insurance plans charge for up to two or three kids and then stop increasing.
I know families with 8+ kids and none have made more than $150k a year and some are way below that.
What I would say, though, is that many women want to have kids later in life, and then this probably has an effect on the number of kids they get to have.
Many people have a "I'll have kids after X" where X keeps changing until suddenly they realize they're running out of time.
It's not the man's job to help with child rearing. It's the exact opposite: it's the woman's job to take care of the house so her man can go out there in the world and build up his business or career (or whatever it is), because that is how he fulfills his role in the family.
That's how it has always been done. This is common to all societies with high fertility.
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.