There's nothing to dispute here–the are obviously factors other than costs that make (some) people prefer to stay put: local family, friends, or even plain old laziness.
That doesn't exclude the cost scenario outline further up, and the relative strength of these effects vary from person to person.
And no argument will "win" such debates for either side. These are complex issues, and even if everyone can agree on the mechanisms at work, you'll get widely different results based on differing importance for the conflicting goals you're trying to optimise for.
Rent control, as shown above, might legitimately add to traffic. But it's also cruel to force elderly people to leave their remaining friends, and their familiar neighbourhood, just because they can no longer afford rent.
Anyone claiming "it's so easy: just..." is invariably wrong. California's legislators aren't stupid. The main contributor to housing problems happens to be the unparalleled economic success of the state, plus a few geographic quirks.
Simplifying to the tired old "free market" vs "regulation" debate is only exchanging one set of problems for another. The US happens to have other locales that dabbled in an ideology-driven "free market" approach. The result are the suburban hell-holes that are Atlanta, or parts of Houston: Low-density clone cities totally reliant on cars, with much longer commutes and huge environmental footprints, almost devoid of street life.
What's usually needed is neither more nor less regulation, but smarter: encourage high-density mixed zoning, create public transport that can compete with cars, encourage companies to adopt telecommuting, flexible hours, distributed workplaces instead of central campus etc...