The evidence for "more housing = slower rising or shrinking housing costs" is so strong in places like Brooklyn and Seattle that it seems almost impossible to really to challenge it.
"so what is the match?" is the natural next question to answer, but the author simply doesn't even acknowledge such a question could exist. cortesoft points to one answer[0]: induced demand. the article hints at underlying land value rising because of the concentration of wealth in city centers is one reason.
like probably many others, i think affordable housing requires a combination of many economic thrusts: upzoning, reducing unnecessary regulatory costs (e.g., reduce bureaucratic friction rather than loosen safety codes), improving mass transit (reduce construction costs here too), mixed-use as default, encouraging small local businesses, last-mile transportation innovations, strengthening labor, progressive tax reform, etc.