it's very tough to say why, but the countries being referenced by this article as having improved the gender gap certainly have lower rates of reproduction. Whose to say good, bad, or why, but the correlation is present.
What about policies that aid the labor force participation of people that already have kids? Subsidized abundant childcare, for example, both lowers the cost of being a parent and makes it easier for parents to go back to work. Like, I get that the _gap reduction_ is correlated with lower birth rates, but presumably we have finer-grained data than that?
I can tell you that better childcare correlates with higher economic output just by looking at my home - both of us work from home and the timing of our child's engagements pretty much dictates when we can schedule meetings.
it's hard to say if that's causative though, because there are so many other confounding factors. wealth also is correlated with fewer kids, and since more working produces more wealth that's also a factor, in addition to lots of other little things.