He's just wrong - in cities that have more building (Austin TX for example) the housing is more affordable.

That doesn't mean 'cheap' if demand is also increasing, but it does cause prices to be lower relative to cities that restrict supply. This is just basic economics, people twist themselves in knots to come up with reasons why the obvious supply/demand is somehow not applicable in this case.

If you allow building in capitalist markets then increased supply will reduces prices as demand is met.

It's not a question of "would it work", the point is that real-estate prices are rising in dozens of countries, and thousands of cities. Zoning is a local issue. Assuming that the reason homes prices are out of control in Santiago, Chile and in Tel Aviv, Israel for the same reasons as in San Fransisco is, well, silly.