No, the problem with housing is that it is a financialized commodity that is engineered to go up in price faster than wages because it's an investment. Not just for individuals, but for real estate companies and banks that gamble with the loans. Zoning laws are a symptom of this, but even if you basically get rid of them (as happens in various places in Texas), the same trend applies.

Those construction companies (really, real estate companies) all get big loans to build those apartments and they do so with an expectation of per-unit profits, often with unrealistic targets unless property values increase even more, and often targeting richer people. When they fail to rent enough at that price point, rather than decreasing rents (which would spook their lenders), they just leave units vacant until they can hit that price point. There are half-empty "luxury apartment" buildings dotting every major city due to this.

The most anyone can point to for the impact of zoning is that prices to rent tend to go up slightly slower.

Your local government is also likely funded by property taxes that are pegged to property values, which is why they never do anything sufficient to handle this issue.