No, that is not the mayor's job at all. Well it shouldn't be. If a mayor tries to make it his job, he'll fail because things like "housing supply" cannot be managed by government.
Want to bring on more high-density housing? Change zoning rules so it is easier to get permits.
Housing regulations tend to be extremely local and I'm not a SF resident, but it is stunningly naive to claim that "housing supply" cannot be influenced by government. I realize Americans love to hate them, but city planning is one thing that governments are actually really helpful to have manage.