I think that building more housing helps, but a lack of supply is not the only cause of high housing prices. Wealth inequality is also a huge driver, possibly the primary driver. If wealth is allocated in a society in a sufficiently lopsided way, then a small number of people can buy up literally all the assets in the society. Regular people are bidding against a small cadre of ultra-wealthy people who can't possibly spend all their money on consuption, so they use it to buy assets like other people's homes. In the aggregate, increasing the housing supply won't bring down housing prices if wealth inequality is also growing above a certain rate.
This can also be fixed by increasing supply. People buy housing as an investment because they expect it to increase in value. The best way to counteract that is to build and continue building so that doesn't happen.
There's only so much land in the places where people want to live. You can't increase the housing supply enough to stop housing from being an asset in a society where wealth is sufficiently unequally distributed. By virtue of its unique scarcity, land (and therefore housing) will always be an asset, and in a society with increasing wealth inequality, the price of assets will always be going up.
