If cities built more housing: 1) Cheaper housing would allow urban families to have more kids. 2) More people would move to bigger cities and be culturally influenced to have fewer kids. Since it takes so much income to raise fertility I expect the latter effect is much bigger!

Since 75+% of Americans and Europeans are already urbanized the main movements wouldn’t be new rural -> urban but rather smaller -> bigger cities. But without NIMBYism big cities could be 2x or more bigger! See Tokyo, Seoul etc. So is NIMBYism significantly raising birth rates?