Housing problem comes from the job market. Cities with good jobs will attract people and the price of renting/buying will go as high as they can bear. Most of the benefit will go to landlords. Moving to a better job market will make you little benefit because your increased pay will be largely swallowed by the cost of housing.

The problem is that employers want their employees physically present at the office, and roads can only carry them so fast, so being closer to job opportunities will reflect on the housing cost. Maybe the solution would be to make workplaces more widespread, improve transportation to expand commute reach under 30-60 minutes, or support more remote work.

They're also mostly having low population growth compared to times when they weren't having a housing problem.

That's variable by country, and slums are no longer acceptable.

The point remains that claiming that everything is down to "not building enough" misses half the equation.

In any case, what is the rationale for building more only to be able to accommodate more immigrants? (Because that's the situation)

because if you took the US without immigrants with its 1.6 birth rate, the economy would collapse

That's kicking the can down the road as population cannot grow forever.

In Europe massive immigration is destroying local cultures for the sake of short term economic gains...