I don’t think the evidence either way is strong enough to call that one a myth. There are lots of other differences between the two countries that could offset the impact of Austria’s childcare subsidies.

There are plenty of longitudinal studies from various geographies, which I would summarize as “childcare subsidies increase birth rates in some contexts, but the effects are complex and depend on program specifics.” E.g. https://pmc.ncbi.nlm.nih.gov/articles/PMC2917182/ and https://clef.uwaterloo.ca/wp-content/uploads/2024/09/CLEF-07...