Are people homeless because they could afford rent, but rather be homeless if they cannot buy? Further, if the rental market was saturated, it wouldn't pay off

Are people homeless because they could afford rent, but rather be homeless if they cannot buy? Further, if the rental market was saturated, it wouldn't pay off anymore to buy rental units. If the demand for rental is just there because there is nothing to buy, it's a clear indicator that there just isn't enough housing (in the desired places). Ultimately the answer will always remain that we need to build more housing. We haven't built enough in decades and it will take a long time to catch up, especially if we don't start on it. We need to relax zoning, remove minimum unit sizes and make it quick and predictable to get approval for new construction. Right now, many of the neighborhoods we love so much would be illegal to build today.