1) Rent-Fixing: This is widespread across the country and the actual reason why inventory increases often (commonly) do not lower prices. There are multiple tactics landlords have used to ratchet up rates without ever letting them drop. These include lease concessions that don't affect the base rent in lieu of rate drops in step with lowered demand, warehousing that artificially limits accessible supply even when new, physical units are hitting the market, and algorithmic price-fixing had allowed landlords within a region to stay in lock step without breaking rank lower. When landlords are able to use such tactics with impunity, they absolutely do warp supply-demand dynamics and allow rent rates to stay high even in the face of expanding actual inventory. For rents to decline, you have to break large landlords' ability to set their price.
2) Builder Subsidies: "Just build more and prices will drop," ignores that the incentive structures municipalities use to draw developers are an additional burden on residents. These for-profit corporations then seek to make a profit on their investment, leading to no direct meaningful inventory increase in the affordable unit range. Worse, developers often target existing affordable units for their redevelopment, destroying the very units that the new inventory is supposed to ultimately reduce demand for. In order for "build more" to drive down rents, it can't be done the way it has been, which mostly ends up being a giveaway to developers at the expense of the tax base and displaced renters.
San Diego is a special case where it seems that the expanding inventory has been driven not by corporate developers, but by the construction of ADUs, which are built by homeowners out-of-pocket, are not tax-subsidized and, in fact, increase the taxable value of property. In this way, they work almost counter to traditional development, and these factors combine with their being created and operated outside of the control of corporate landlords means that they represent new and meaningful competition to those landlords. Thus, efficiencies are sought and prices drop.
Encourage ADUs. Avoid subsidizing large developers and corporate landlords. If there is the political will, get the government directly involved in constructing subsidized owner-occupied, human-scale housing, as in Singapore and the pre-Thatcher UK. Relax zoning and build out public transit and car-free infrastructure in order to reduce the accessibility premium, as in Japan. That is how "build more" actually can work to lower rates.