> Is our goal to reduce the gap between the rich and the poor, or to improve the lives of the poor? Because the gap itself is actually irrelevant to poor people's lives.

The resources to improve the lives of the poor (and not just the very poor but the struggling middle classes), have to come from somewhere. They are not infinite. So unless you're creating a bunch of wealth from nothing, then it naturally requires a more equal distribution of existing wealth.

> China which has been consistently overbuilding housing for decades now. They still have housing issues.

This is in fact a problem of unequal distribution of resources: one of the main reasons there are housing issues is because houses have been seen as an investment and gobbled up by those who can afford to do so (all my friends in Beijing own multiple apartments), so prices rise and the poor can't get them anyway.