For housing maybe. It’s useful to have governments nudge developers to build affordable housing, which is less profitable, but if you have enough supply it can work. It does not work in most of Europe, where land is scarce and expensive and developers still want money. More than zoning laws, housing issues in Europe is in large part caused by the lack of government-build (or subsidised) affordable housing on the low end.
For healthcare, hell no! A single payer brings massive economies of scale and a lot of bargaining power, which limits price gouging. Hospitals are local natural monopolies, it makes no economic sense to have enough of them around to have meaningful competition. Demand is very inelastic and people just pay what they must to get treated (when they can pay). Insurance companies have interests that are directly opposed to those of their customers. Most people do not cost much for most of their lives, but have crippling expenses at some unpredictable points when they get sick or have an accident. National social security schemes smooth out the risks over the whole population, which makes everything more manageable. To me, healthcare is the opposite of a situation where free market makes sense.