Doesn't help the countries losing people. If demographic shrinkage is going to be a problem, it'll be an international security (as in, avoidance of conflict) problem. We need solutions that don't involve robbing other countries of their population, and particularly skilled parts of their population, even if we think that for now they have a "surplus" of people. While overall it's probably a net gain (skilled industrious people generally achieve greater lifetime potential and output in prosperous, relatively free countries), the gaining and losing nations are winners and losers, respectively.
Families with 2 or more children were common when I grew up (many with 3 or 4), when "family planning" had been well established, so birth control can't be entirely blamed. What's changed? Most of the basic elements of a middle class life are relatively cheaper. An outlier is housing. Transferring money to individuals using other excuses is just a band-aid. We need to fix the underlying problem, and if we get it fixed, we'd need to remove the band-aid so we can fix other problems.