One key thing to understand is that private companies build homes, not governments. And if it's not profitable to do so, it will simply stop. And if not at least 75% of the project is sold beforehand, the project is stopped altogether.
And this is how our government's ambitious plan to build ~100K new homes per year got crushed. In particular due to rate hikes, making mortgages even more expensive than before, so potential buyers opt-out.
This part is critically important to reflect on as it keeps happening. There's an economic downturn and the building of new homes slows down or stops. This happened over here from 2008-2014 creating a massive backlog, in turn leading to a pricing boom in recent years. Ideally, you'd keep building during a downturn but nobody knows how as there's no buyers and it's too big to subsidize. Hence, at least in the Netherlands, unaffordable prices are not due to zoning, they are a supply deficit based on downturns.
There's also labor shortage, material shortage/inflation, far more complicated construction due to the energy transition. And yes, higher than usual immigration is a factor. One of many factors.
And guess what? In my area, the grid is full. A few more homes can get connected but not a single new company can connect for the coming 10 years or so and it takes billions in investments to fix. This in reference to new housing supply having lots of other dependencies.
We also tried rent control on private rent. Guess what? The land lords simply sell the property, now you'll have even less rent supply. Well done.
My point being: increasing supply is not easy in this perfect storm. Surely the entire western world would not have this problem if it was a simple as a policy change. It very much involves constraints in the physical and economical world.