You don’t. Correlation doesn’t equal causation, of course. My point was that GPs solution might not be enough, not that it was unviable. I tried to demonstrate that by showing that in the worst case it could potentially backfire, not that it necessarily will. I was painting in pretty broad strokes though, mostly because I don’t have any better ideas, so I could see how it just comes across as FUD.
Not to sound snide, but I also don't think it's as straightforward as linking two correlative effects. High grades in school is also often predictive of economic background, race, gender etc... that themselves are correlated to birth rates. I don't see here much reason to believe that providing early childhood care would decrease birth rates.