Profit motive leads to more shelter being built. The same reason why you want food production for-profit.
Not everything works best with profit motive (eg, healthcare) but if you want affordable housing then maximizing profit motive will help produce more housing.
No, profit motive leads to more shelter being built for those who can afford it and everyone outside of that gets to live on the streets. See: Seattle.
> Well, we had food and shelter before we had profit motives
Did we? The Hobbesian original condition is one of the strongest "profit" motives around. Natural selection is unsentimental and unforgiving. This idea of distributing things according to "need" is a very recent novelty.