The vast majority of the problem is bad (i.e. density-restricting) zoning, not bad bike/ped/transit infrastructure itself. The latter is a symptom of the former.

At least in my city this isn't really true. It's an illusion of the cost of housing typically being measured by cost per square foot. But when you compare cost per unit, it ends up being similar. In fact if you reduce your vehicle costs by using other transit modes, you may even save money. However, this does require knowledge of how to travel car free, which not everyone has. It also may require some longer travel times on certain errands which can't be done in the urban core. But this is usually an occasional inconvenience vs the long daily commute, so you still save time.

However this is just my experience and it may vary by city. Also I was comparing middle class areas, if you are willing the live in areas with high crime/poverty you might save more money.