I think your question is putting the cart before the horse in asserting that the "normal" expectation would be for the birth rate to rise because daycare is subsidized. That's a claim that needs verification. Asking us to (dis)prove the negative is not the correct approach here.
People say "I'll have more kids if I have free childcare" on surveys
I think this is a misrepresentation of a more correct "I'll be more inclined to have kids if I have free childcare". The latter does not attest to an increase in the amount of children, only a more eager approach to deciding to have them.
If you consider that people tend to stabilize financially as they age; the effect might merely be a matter of what age you have your children, rather than what number of children you end up having.
Secondly, the assumption that removing a restriction therefore triggers something is a logical fallacy.
My car won't start. The fuel tank is empty. I fill up the fuel tank. Will the car now start?
You cannot judge this accurately because you have not considered if there are any other blockers to the problem that you have not yet addressed. Your question is rooted on an assumption that the cost of childcare is the only reason people are not having the children that they want to have, and you've provided no evidence to support that claim.
Thirdly, any assumption that the amount of children that are born is therefore indicative of the amount of children that people wanted to have is affected by people who have children by accident or who do not engage in (accurate) pre-emptive family planning.
You cannot look at the amount of children being born and use this as a measure of how many children people want, nor how people's plans with regards to the amount of children they want to have might change over time.
Fourthly, you claim the birth rate is "low". Compared to what?
The only surefire comparison to make would be a comparison between the same culture/country/... either having or not having subsidized childcare. It seems like you're assuming an equality in birth rate between different cultures/countries/...
While there are some reasonable comparisons that could be made when you account for the variables, I'm not convinced that you've accounted for the differences between different cultures/countries/... enough that you can conclude that one specific key concern (childcare subsidization) lies at the heart of a difference in birth rates between the different cultures.
To put it differently, just because my neighbor earns more than me does not necessarily mean that he has (or should have) a car that costs more. There may be myriad life circumstances that lead to me and my neighbor spending our money differently, and it's not just related to income.
Similarly, who says that the Nordic countries don't just start from a significantly lower birth rate than what you're comparing it to if you assumed neither had subsidized childcare, and the subsidization actually does increase the birth rate, just not particularly in a way that it overtakes the other region you're trying to compare it to?
Overall, I think your premise is flawed. It's built on an assumption that is not backed up by any evidence that I've seen, it oversimplifies the problem space by omitting several key concerns that are not at all addressed by childcare subsidization (or lack thereof), and it uses a metric that is at best inaccurate.