Well, I totally agree with you but in MLB's case it's part of the CBA (and still technically an improvement over the US at large, where it's still not even mandated). I seem to recall players can extend it too, though they might not get paid (i.e. they'd be moved to restricted list). Which is terrible of course. But I think in this case it is the culture of pro sports, rather than American culture - even if they are both rather patriarchal - that makes it extremely unlikely for a Dad to take two weeks off, let alone 12 (which some employees will allow!).
Fans also have the often truly bad take of, "why didn't they time the baby so it wasn't during the season?" Which doesn't really even make sense, biologically or mathematically, when you consider the times of year players actually get to spend a lot of time with their wives. In this case it seems like the Castellanos may have actually been trying for an off-season birth!
End thread drift 
