With the creation of the birth control pill the state (and men) lost the ability to coerce women into having children.
From my experience women naturally want to have children, and not just the women of my generation, but the children of my daughter's generation as well. Hormones can destroy all your best laid plans, and I say this as a woman who wanted to play pirates and explorers as a child, not "house" with baby dolls. When it hits, and hit it does, "Baby fever", even when one has had one or two, is very real. When you pick up a six month old infant and it snuggles next to your breast, the hormones start to pump out.
I just had to resist it, not for financial considerations, but because doing it the way I did it required all my reserves of energy, and more importantly, patience. That would have been true whether or not I was working full time, although I was, in fact. My husband never much cared whether we had children or not. He agreed more for me, I think, that is until our son was born. He started wanting a second when our son was barely a year old. We sensibly waited a few years.
Had we lived in a different time he would have wanted a house full, but then of course he wouldn't have to bear and birth and nurse them, and stay home with them all day.
If the state wishes to encourage larger families they could start by providing free child care, perhaps at the place of work itself, and a stipend per child.
However, my suspicion is that the change would be modest. Most women with any sense would prefer to have a man with a commitment to both her and their children, and many young men would prefer to have the sex with no commitment and with lots of disposable income for their toys. Imo modern society's perpetual adolescent male seems to be a very real phenomenon. By the time a woman finds a man with whom to have children she's in her 30s and her child-bearing years are limited. That's true for lots of "career-women" as well. They've been sold a bill of goods that you can have it all, and then they discover that it takes until you're at least thirty to establish yourself in a career, and by that point if you don't have a committed man in your life, having a child is almost an impossibility, and even if you do, your ability to get pregnant is really hampered,which is part of the reason why infertility treatments are such a lucrative business. A lot of those good eggs are just gone.
As for Italy and apparently some other countries in Europe, the one child thing seems bizarre to most Americans. There are three child families and two child families, but I don't think I know any families who could have children who have only one. It's considered bad for the child for numerous reasons, and I agree with that.