Population booms in good times hence the boomers during the post-war economic expansion. It isn't a Children of Men scenario. Parenting is one of if not the biggest economic decision people face in their lives.

No single issue like free child care will change things. Unless we see real wage growth again and better affordability of housing and utilities we won't see higher birth rates. It is a huge sacrifice for many people and turns very stressful in a bad economy.