You really can build enough housing to lower prices. Tokyo has. You can get a small Tokyo studio apartment for $700 per month. [1]
This is because they build housing. San Francisco, London, New York, and Seattle all don't build housing and prices go up. [2]
If housing were easier to build people would build more affordable housing. No developer cares if they build expensive housing or affordable housing, they are maximizing return on investment. Turning a 1 story house into a 3 story house for a 30% return makes developers just as happy as developing 20 luxury condos for a 30% return.
And even those luxury condos provide housing for people who would otherwise buy old condos and renovate them. As long as the housing is more dense and there is more of it, prices will come down.
[1] https://resources.realestate.co.jp/buy/average-rent-in-japan...
[2] https://www.ft.com/content/023562e2-54a6-11e6-befd-2fc0c26b3...