Zoning and parking minimums are the real problem. Ensure houses are safe (fire, collapse, etc.) and let the market decide where, when and what should be built.
Exactly -- Tokyo has kept housing costs under control for this reason.
There are zoning codes in Tokyo, but they're quite flexible compared to what people in the US are used to. In addition, perhaps most importantly, it's not possible to oppose construction, as opposed to the the US, where literally anyone can halt a construction project indefinitely for literally any bogus reason. "It might cast shade on 1 square foot of the street!"
It was a sad day when I learned there exist actual shade inspections as part of the development approval process in some places. I legit thought it was just a way to mock NIMBYs who say that stuff. It was on that day I kind of just stopped hoping for a better future in terms of housing.
They are at best a very minor cost addition when it comes to large projects. I see more problems from insufficient parking which is why many of these exist in the first place. There is very little independent research into the topic, it's mostly from the anticar crowd.
I feel like immigration is effectively unlimited from the perspective of big cities. There's friction moving states, and some things keep people anchored, but the market is nationwide. If you built enough in NYC to make rent comparable to other major cities, then significantly more people would want to move to NYC.
From the perspective of any one big city, yes. From the perspective of, say, the top 20 cities in the country put together? No - not unless immigration into the country is unlimited.
You can't build faster than demand is rising (because building increases demand again). Except if demand goes down due to an external crisis. Or if you magically could double housing with a snap, but once again later once that new housing gets occupied... more density means more job means more opportunities.
"Just build more" is not a solution on its own, at all.
(Or, we could in most cities. A few, like New York and San Francisco, are very limited in terms of available land.)