> It’d also be conducive to his wallet if they get born before Jan 1st.
Agreed. Also, being born even on December 31 2019 qualifies you for having another dependent in your household for the entire 2019 taxes, so it's a double whammy.
A due date after September-ish ensures you're only paying the medical treatments/screens/hospitalization/etc for that year and that year only (vs, as you mention, a due date that happens potentially anytime between Jan-August).