Why not both? Subsidize development and permitting fees with vacancy taxes.
In fact, those subsidies are often part of the problem. The only way to get new development approved in many cases is to include a number of below-market units in the development… even though they are subsidized, that mandate kills the economics of the project. It happens often enough that you start to wonder if maybe that isn’t the point.
It's entirely correct to say that cities are the main obstacle to building housing, but it's not true to say that everyone wants to build more.
Whether or not a project pencils out depends on local demand and interest rates. An unfortunately common sight in SF is that developers start construction before getting all the required permits to finish, then the city drags its feet for half a decade, then when the permits are finally granted the building no longer pencils and the foundation or skeleton are left to rot.
The timeliness is important; the goal for most finished buildings is that they provide so much supply that building new ones isn't financially viable.