The majority of the "more housing = cheaper rent" success stories crowed about on the internet correspond with net population exodus. Austin, Minneapolis, now San Diego.
Edit: I think we should build several million more units of housing in the US. I'm salty because all the new housing I've seen is ugly shitboxes owned by national property firms that make everywhere feel like nowhere.
Do you have a source for the claim that the population of Austin has decreased recently?
Austin does not appear to have undergone a "net population exodus", at least not one visible on any of the charts I found (I stopped after FRED).