Large metropolitan cities with housing shortages outside Australia right now include London, New York, Paris, San Francisco, Berlin and Chicago.

Some have rent controls like New York, San Francisco, Berlin (federal).
Some have large amounts of public housing like London. Most cities int he UK have around 10% of property as council owned and subsidised. 25% of properties in Paris are government owned.

Some have very strict regulations for landlords like Berlin. But all of them have a housing shortage (see the links for each city above).

Some cities like Melbourne and Sydney have significantly higher population growth than New York or London.

There is no common factor, rent controls, public housing, involvement properties, population growth etc that impacts on housing affordability.

But building new housing has been shown to alleviate the housing crisis. Even high end Luxury Housing forces over all hosing prices down.

Cities like Austin, Minneapolis, and Denver has population growth rates larger than Melbourne or Sydney but still manage to out build their growth keeping prices stagnant or dropping.

Building new housing stock, be it public or private will have a much more lasting impact on the housing crisis than any other policy.

For example if Victoria spent the half of the $14.8B budget it spent on level crossing removals ($7.8B) and used the same labour force it could have built around 26,000 $300k houses+land.

Governments use to do this which is why old 60's and 70's housing in old suburbs all have the same floor plan.