Build more housing? In a place like Austin, you can just keep building out, basically. To a point. Eventually cities doing this reach a limit. Houston and Atlanta are pretty much at or beyond that limit.
And it's not that building low-density SFH housing is the most economic. It's simply the most subsidized. Every road, every parking space, every sewer pipe, every water pipe, every utility pole, every school, every hospital, every police station, every fire station... they all add factor in to the true cost of housing and the more spread out things are, the more expensive those things become. Taken to extremes, look at the billions Houston spends now to add just one more lane (because this one will totally solve traffic) on, say, the Katy Freeway or the ring roads.
Yes it does need to be affordable. NYC is the posterchild for this. Nothing that's getting built on billionaire's row will ever trickle down to being affordable housing. They build ultra-luxury housing because it's the most profitable and it does absolutely nothing for anyone else because these units are just ways for non-residents (mostly) to park wealth and not pay their fair share of taxes.
Rent control is the wrong solution for the right problem and it's typically American. By that I mean it forces the solution onto private landlords who are going to do everything possible to get out of those obligations and deliver subpar but compliant housing. And they'll demand tax breaks for it. When in fact the solution is for the government to supply a large chunk of the housing market ie social housing. But there's a pervasive and wrong idea that we can only solve problems in the private sector and that's nothing more than a wealth transfer from the government to the already-wealthy.
"Just build more housing". Yeah, and then you get Houston. Cities need to be planned. Cities require infrastructure. And one of the most important thing cities need is public transit infrastructure, something sadly lacking in virtually every American city.
The core to so many of these problems is that we need to stop treating housing as a speculative asset. Owning two or more houses should be incredibly difficult and expensive and should be taxed punitively. By this I mean the capital gains on non-primary residences should be 80% and property taxes should be significantly higher.