You mean like the USA? Or just about every developed nation on the planet?
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.
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)
In Europe massive immigration is destroying local cultures for the sake of short term economic gains...
European governments do not want to face this reality and prefer to try to convince people that immigration is good (polls show that the people increasingly disagree).
Case in point: Tokyo. You just need to make sure you _have_ to live in The City, and that there are no other options for you. And a shrinking population makes that _easier_.
In some European countries population growth is only due to immigration. In Spain almost 19% of the population is foreign-born!