Who's checking to see who is born, who dies and where the money goes?

I would expect an increase in reported infant births (and reductions in reported infant deaths) if someone offered $1000 no-strings-attached! In fact, I'd expect a radical jump in births. And I'd expect researchers to skim off a proportion of the winnings too.

Yikes, your reaction to a study showing infant deaths halving in Kenya is to assume the researchers skimmed some of the money?

The article describes a randomised controlled trial, and explains the result is from better access to healthcare. What would make you happy, should they have sent you the money instead?