If, say, gold was discovered in Honduras and the nation became rich and immigrants started flocking to Honduras because an uneducated person could feed his family there, then the lack of white people having kids would be something the media would be SCREAMING about.

I've been trying to come up with a similar analogy.

I wouldn't even say they'd be screaming about it - I bet if people having kids suddenly became immensely profitable, they'd flip a switch on what we see in the media and within a couple of months women would suddenly be horny and aggressively seeking out men to have kids with.

There's something in humans that cause/diminishes desire for sex based on messages from the social group. Probably tribes of people survived who didn't have kids when famine was coming and did have kids when prosperity was on the horizon.

If it was beneficial for themselves for people to have kids I'd bet they know exactly how to change the messaging to accomplish that.