Because housing is a vital human necessity, so much like healthcare, the price ceiling for it is effectively infinite, because almost no-one can reasonably opt-out of purchasing it. (And due to how it's financed, it's not even really capped by people's income, since all of society is built with the requirement that everyone spend more than they could afford, just to get basic essential housing)
Saying "new (for-profit rental) housing construction will fill the shortage" is like saying, "Yes, Nestle charges an obscene amount for our water, but if we just let Nestle pump more water from the ground, they'll voluntarily sell that water to us for less".
The bottled water market has exactly 0 things in common with the housing market.
Every city is growing and adding more people than new housing construction. Thus rents go up. Hm, wonder what the solution would be? If a landlord has a unit open for 2 months maybe they would drop the rent? Crazy idea.
It doesn’t have to be totally public, just publicly provided supply with eventual private ownership, as is the case in Singapore. The problem is that the USA is a country (many cities/towns to live in, some more desirable than others), while Singapore is a city state.
Saying "new (for-profit rental) housing construction will fill the shortage" is like saying, "Yes, Nestle charges an obscene amount for our water, but if we just let Nestle pump more water from the ground, they'll voluntarily sell that water to us for less".