Housing isn't the sole factor in declining birthrates. Every industrialized nation is going through the same phenomenon and not all of them are having housing crises. It's a combination of various associated costs to raising children (housing, early childcare, food, schooling, extracurriculars, hobbies, etc.) that have risen faster than most people's wages. The nuclear family model also significantly diminished the role of parents and other senior relatives in early childcare which would've created significant savings in terms of both time and money.

Also, living without children or even a partner at all has become significantly more viable in the past 40+ years or so. It wasn't until 1974 when women in the United States could unconditionally open bank accounts out of their free will: https://en.wikipedia.org/wiki/Equal_Credit_Opportunity_Act. To not marry and have children would've practically been a form of ostracization. So many people, especially women, "settled" into marriage and family life simply because there was no other option. I think the decline in birth rates is a fair tradeoff to forcing fewer people into a lifestyle they did not want which would inevitably result in broken families and broken kids. What society needs to figure out is how to decouple productivity from raw human headcount, as well as how to avert potential humanitarian disasters that may occur when the decoupling happens.