I suppose it depends by what you mean by “high birth rates” but Israel has a high cost of living, Tel Aviv is comparable to most American superstar cities when you account for resident’s lower incomes, but the country has a birth rate well above replacement, at roughly 3 births per woman.
Of course, Israel also has a nearly identical rate of secondary and tertiary educational attainment to other OECD countries so I’m not sure that makes the argument against education.
High cost of living can happen for two reasons, either everything is expensive, or you make no money. And guess what, people with no money have way more kids than people with a lot of money in middle class lives in developed countries, who then say lack of money is why they don't have kids.
The one thing you can do to stop teenage pregnancies is for example to educate young girls, give them menstruation support and allow them to go to school. This effect dominates vs "oh I can't put my kid in private school" that we say is the reason for no kids.