I don’t think that’s strictly true. Flavela housing is also smallish but that doesn’t stop reproduction. It’s more a cultural thing with many other contributing factors, like economic outlook, cost of living, etc.
It's not housing. It's basically lack of time. Japanese are allowed to work almost all of their waking hours. So they do. They pretty much have to because rent is on average always as high as people can afford. Not much time for anything else.