American women say they want closer to three kids than two kids: https://www.nytimes.com/2018/02/13/upshot/american-fertility... (average 2.7). Half of women with kids, and even 30% of women overall with a college degree, would prefer a homemaker role to working: https://news.gallup.com/poll/267737/record-high-women-prefer.... These women would probably have more kids if it was easier to have a single income household.
I suspect “logistics” has more to do with it than “independence.” Our education system and economy, in particular, causes people to spread out and move away from family. Raising kids without family around is much, much more difficult.
There is also the fact that the feminist revolution stalled in this weird place where women have freedom to pursue careers--but only if they can shoehorn themselves into existing roles designed for men. Want to grind through your 20s in Big Tech or Big Finance seeking promotions? Fine. Want to take time off to have kids at an age when it's the lowest risk and you have the most energy for it? Then you're off the career ladder forever! That's not some unavoidable consequence of "women's independence." Our retirement and pension systems are very much designed around the biological reality of men's bodies working strenuous industrial jobs. But our career ladders haven't changed one bit to accommodate the biological realities of motherhood.