Be careful what you wish for. China also has the hukou system, which is sort of an internal passport that effectively prevents many people from moving into the popular tourist locations. Get out into the rural areas and you'll still see a lot of real poverty, although housing is cheap enough there that there aren't many homeless.

There's also a cultural factor at work. Allowing a relative to be homeless causes loss of face so family members feel more obligated to pitch in and help them out, sometimes to the extent of providing a free room. (I'm stereotyping a bit here but it's generally true.)