The population growth is basically a transitional state between high infant mortality / high birthrate to low infant mortality / low birthrate. There's no society in existence that had sustained "high birthrate / low infant mortality".

Play the following graph from 1800 to today:

https://www.gapminder.org/tools/#$state$marker$axis_x$which=...

It's very clear where we're headed and we're actually closer to the stable point, given China and India are already near the birthrate of 2.

> ... are still happily multiplying in an unsustainable way.

See the graph above. This is already not true.

> Unless we manage to export this kind of progress

This has already happened mostly. Africa is the laggard currently but is already rapidly catching up, and it's crystal clear that we'll get there.

I recommend taking the gapminder.org test. I think you'll be surprised how wrong you are.