https://www.youtube.com/watch?v=TaU1UH_3B5k
Not just scarcity, but the lack of mixed use development, the lack of high efficiency city logistics (moving people, people's belongings, food/freight/cargo to shops/restaurants and bigger stores) prevent most US cities from being adequately flexible enough to adapt to changing economic patterns. (And to allow innovating new patterns, etc.) Not to mention that most metro areas are a zigzag patchwork of planning jurisdictions, and all of these municipalities are incentivized to push certain stuff away from them.
If that’s true (I’m not from Houston!), then it does seem like zoning reform is probably not enough to solve America’s housing issues.
The problem is just that Texas, as a whole, has a planning philosophy of outwards, not upwards. Things are densifying in job rich areas now that they've just about expanded as far as they can (the city proper 669 square miles, or about twice the 5 boroughs of NYC, twice the geographic size of its sibling city Dallas, and 150+ square miles larger than Los Angeles; none of these sizes include surrounding suburbs, by the way).
All of this is long way to say that you're right – it's not just about zoning. It's equally about ending the grip of car-centric planning and not just allowing, but incentivizing densification and walkability. The fact 2/3rds of Houston is in (increasingly-frequent, due in part to the sprawl itself not allowing anywhere for water to runoff to) flood zones, too, is apparently not incentive enough.
[1] https://s.hdnux.com/photos/51/06/32/10774164/4/1200x0.jpg [2] https://s.hdnux.com/photos/51/07/70/10780179/4/1200x0.jpg [3] https://s.hdnux.com/photos/51/07/64/10779939/4/rawImage.jpg
The problem is that there's an enormous missing middle (and high) density development regime in the US that would allow cities to adequately track the changes in housing needs.