A key to this strategy is
building sufficient numbers of housing units; if you split these between units to be offered in the market (prevention) and units dedicated to permanent housing of the currently unhoused (cure) you bring down costs for people with income seeking housing in the market while providing immediate (as the units become ready, obviously there is a lag from adopting the approach as policy unless you have vacant capacity that can be instantly repurposed) assistance to those who even with greater supply are not inmediately able to make market rents.
You can't execute a Housing First strategy effectively without adequate housing supply, which is the most fundamental problem in a number of locales, including the Bay Area. But additional market supply alone is not sufficient to address the urgent homelessness problem.
> IMO govn't need to relax the regulation to build more houses and drive the cost down
That absolutely needs to happen, and that helps with prevention, but except for the fairly-well-employed homeless (a group that actually exists and is often ignored, but isn't a big part of the homeless problem), adding new market rate supply alone does not provide significant assistance to the currently homeless.