I've looked into moving to Japan. My family has a house in Setagaya, which had severely distorted what I thought Tokyo living to be. It turns out, a typical one bedroom apartment is like 20 sq meters.

Separate from that, I don't believe zoning has anything to do with housing shortages. Housing has "induced demand" similar to traffic. You can look at heavy traffic over a bridge and think, "if only we had another bridge, that would alleviate the traffic." But it turns out, when you do that, more people end up using both bridges and traffic becomes as bad as it was before. Housing is similar. If you build more houses, even more people want them. Look at places like NYC, which has extremely dense housing. It's still expensive.