(The comment may indeed be true--sort of by definition to at least some degree--for new construction, although getting that access can be pretty straightforward, to the degree anything involving construction is straightforward. It's more that housing prices are an issue in popular places where a lot of people want to live. And, yes, vibrant college towns fall into that category although not to the same degree as elite cities.)
I just don't think the conspiracy that the OP postulated, "We all want our properties to increase in value so we collectively conspire to suppress supply" is a very plausible explanation when applied in general to the entire country. For example, again in the Madison area, most of the home building is occurring not in the city proper but rather out in the suburbs and exurbs, which are independently governed. Indeed, in a fast-growing suburb, the property values can actually increase as homes are built and the town becomes a more desirable place to live.
When the OP said "everyone wants to live in the same few places", I didn't interpret that to mean specifically NYC/SF, because that's plainly false. I interpreted it to mean urban vs. rural/undeveloped areas in general.
But the problem is that home builder cartels exist all across the country, and they own a lot of the desirable land around the urban areas. For example, in my own area of Madison, Wisconsin, we have a company Veridian Homes that owns giant tracts of land all around the outskirks of the Madison area. They're the dominant home builder here.