I had a kid recently and the entire process of labour was free. We even stayed an extra couple of nights. We had a midwife with us the whole way through the pregnancy, and we still didn't pay a cent.
As for leave, I got six weeks at full pay; my partner gets a year (approx 60% at full time pay, 30% at minimum wage, 10% unpaid - that's how she organised it).
The poor pay nothing, the lower middle tend to get stuck with a bill that they can end up affording, and the upper middle and above often have insurance plans that give them great care for an out of pocket that they don't really notice (this is a huge generalization and actually varies )
As for leave many Americans take pride in not taking time off. One lady i met was saying that a recent round of maternal leave increases were unnecessary as she was able to go back to work the week after...
I'm definitely in favor of having kids because I'm not sure what sort of end-game a civilization is aiming for if they decide to stop having kids, but I've yet to have a discussion on this topic with a younger native European that actually wants or has kids. The answer is always either "well I want to be able to travel and have fun without caring for a child", or "the world sucks so I don't want to bring a kid into it".
I was finding that a life of travel, hobbies, and low responsibility was becoming more hollow to me.
And the world is a tough place, but it always has been and always will be. I just didn't think that allowed me to absolve myself of responsibility in making it better and doing what a saw was my duty.
Also kids are pretty cool. They're like little acid trippers. I didn't want to miss out on the chance to have that experience.