Same things go for schools, food in schools, universities, et.c., they are all "free", as in paid for by everyone in proportion to their wealth, instead of individually by the parent.
It isn't cheaper to raise kids. They're actually losing more under the socialised system because they still have to give up enough resources to raise the kids and now also to cover inefficiencies in the government program. On average everyone would be worse off. Or you're going to be taxing grandma so instead of her providing helpful childcare services she has to go and work at McDonalds to make up the difference or something. Society is geared towards supporting children already - routing the resources through the state or through the family can't make it easier. Men work to support families, women work to support families, grandparents have nothing better to do than support families. The resources were already tagged to support families. The government can't magic more resources into existence if raising kids is expensive. They'd be taxing families so instead of using the resources to support their families, the resources are being ... handed out to support families. There are no gains, so it isn't going to help.
State spending can reallocate; I can see some potential success in using childcare to free up high performing parents. But if the problem is median experience, reallocation can't do much.
Postscript The taxes in these countries are no joke. Young families are losing a lot more to the tax system than the "free" childcare makes up for. If you traded the free childcare for paid childcare and low taxes, they'd likely be in a much better position to have kids. [Taxes] approx. = [Cost of Services] + [Admin Costs] + [Cost of Rorts]. That isn't making it easier to fund a family.
In actual practice there will be particularly fertile women that have more children and their childcare needs will be draining resources from others, making it harder for them to form families.
I should stress that I don't think childcare costs are the big driver of low fertility. But to say that resource reallocation is going to help is a bit ... I mean, where are these resources going to come from? Everyone was already putting their resources towards their families, diverting money away from those families is the engine that powers the state. That is where the soldiers come from, the welfare comes from, all the labour, where the goods get produced, etc. We'd be asking them to stop caring for their families and instead spend all day caring for their families but according to how the government wants instead of what they actually think they need. You can't take from them, give it back to them and expect them to have more. They aren't even going to have the same.