Building houses isn't free, and when there's a huge demand for land to build on, and all the land is already taken, prices to build the house jack up, and everything follows. Find me a single city anywhere with similar growth that didn't have housing and rental cost go up and maybe I'll reconsider. But right now, I stand by the most simple explanation as likely the cause.
The most simple and obvious explanation by far is that demand has outpaced supply. The thing your analysis is missing is that this mismatch is entirely artificial. It is more or less illegal to build new housing. It is structurally and legally impossible to construct enough housing to meet supply. Zoning prevents most of it and excessive regulation takes care of the rest. The cost of housing is astronomical because it's illegal to build enough of it.
I'll turn your phrasing around and say show me a city that makes it legal to build as much housing as people desire for 30 years that still has high prices and I'll consider your point of view. But until then, it seems to me that you're going to have to make it legal to build housing if you want more housing.
