In Soviet sphere of influence in Europe there was sudden drop in fertility rate after fall of communism. The fall of communism meant that we went from empty shops and lack of opportunities to full shops and plenty of opportunities. So exact opposite happened to what you predict.
Additionally, recently e.g. Polish government introduced very generous child support in a program known as 500+. There is no noticeable change in fertility rate.
IMO the biggest reason why women decide to not have children is the perceived opportunity cost. Loosing on studies, career and freedom of being young. Communist countries did not provide young women with opportunities and did not demand high competitiveness, so perceived opportunity cost of having children was a lot lower.