The reason there is a housing problem is that you cannot build houses as quickly as people can (and do) arrive by plane and settle permanently in a country. It is primarily, though not entirely, an immigration problem.
Giving you Spain as an example, that I am more familiar with: We've had an insane housing crisis for the last decade, and yet we have about 4 million empty houses across the country.
The housing crisis has nothing to do with immigration and everything to do with an unregulated basic good. Having a roof over your head, should be a basic right, and no-one should be allowed to speculate and make crazy money out of your need for it.
The prices rise because there is no regulation at all and it is one of most lucrative forms of investment, with a lot less risk than other forms of financial investment.
The vast majority of people in the UK, for example, have long commutes by train. I have workmates that have 2.5 hours of commute each way, and while that is extreme, 45m to an hour is completely normal.
Translate that to Spain and your radius for an available house is immense.
There are 266 municipalities around Valencia, each municipality with several (some dozens) of towns, that you can reach by car in less than 60 minutes. In Madrid, there are 179 municipalities less than 60m away from the city.
Again, it would be fantastic if you could walk to work. But that is a massive privilege in large cities.
Correct me if I'm wrong, but I believe that the social organisation of Italy won't be massively different, i.e, lots of small towns around a major city.
I still stand by my point. The housing crisis has nothing to do with immigration, and everything to do with rich people and corporations being allowed to speculate with a basic good.
Paraphrasing Gary Stevenson, if you allow wealth accumulation past a certain level where the wealth cannot be spent in normal things, the uberrich will start buying ALL the space. And that is exactly what we are living now.