I have a blog on it but not linking it here might be bad etiquette. You can start with Claudia Goldin's paper "he Downside of Fertility" where she explains that women's education is only correlated with lower fertility when patriarchy tries to stop them from benefiting from that education.
The more that men can credibly signal they will be dependable “dads” and not disappointing “duds,” the higher will be the birthrate in the face of greater female agency.
Patriarchal men don't actually desire non working trad wife spouses in today's economy. They are after double income with their wife having double the work - at her career and at home. For instance women without college degrees have much lower marriage rates in America.
I'm not saying patriarchy wouldn't love to have a 1950s single income economy but that economy is an aberration in the history of patriarchal societies. What was common was preventing women's education but that only worked as long as your uneducated wife could still run a shop or hunt for you (see Philippa Gregory's Normal Women).
Participation in a modern economy requires an education and the job you get with an education is not compliant with long hours of work at home.