Not necessarily. It's possible that no amount of money would solve this problem. Birth control inherently broke the previously built evolutionary mechanism that insured that the extremely strong built in desire for sex would result in kids. That's no longer the case, and a lot of people would decide to not have kids even if money were no object.
As you point out, Finland famously has incredible family support, and also a birth rate under 1.3.
This sounds extremely plausible to me, but I would be very careful about conclusions from such studies, because I believe the general expectations of society as a whole regarding child-raising matter a lot and you can't easily quantify that.
Anecdotally, when my grandmother did not birth a child for two consecutive years in her thirties the village priest came to investigate (!!). Expectations have shifted massively since, and the single/dink lifestyle is way more "acceptable" now.
As you point out, Finland famously has incredible family support, and also a birth rate under 1.3.