Well that's the problem, isn't it? The self-correction is for nature to destroy us so that something else can take our place.
It's completely possible for births to be below the population replacement rate until the people die out. It already is in first world countries save for immigration, and that can't save you if the third world industrializes and automates to the point they're in the same situation.
If we don't want that to happen we may have to start finding ways to enable people to marry and have more children.