I live in Japan and have Japanese friends. Lots of Japanese I know want to have kids, and wouldn't mind having 3 or more kids. They love kids! And it's still the dream of many women here to get married, have kids, and be a housewife since they know how terrible the careers are in this country.
But 1 or 2 is the limit due to work/time, finances, practicality (housing, vehicles/public transit). The families I know with more kids are the ones who can afford having a housewife or who have multi-generational homes (live-in grandparents).
What I'm trying to say is that I think one major reason people aren't having more kids isn't because they don't want them, but because society isn't built to enable them to.