That doesn't help Italy though. People get the citizenship and move on.

You can't think of citizenship in the normal way. You have to think of it within the context of the Schengen. Those people even have an easier time moving on out of the Schengen to, say, Canada or the US.

Of course it does. Labor increases production and therefore GDP. If people are leaving war torn or destitute parts of the world and finding available work (perhaps from the government if not private industry) and housing made available and its in supply owing to a shrinking population, social safety nets for once, etc, where would they move to? Germany to spend more for basically the same set of circumstances you'd have living in Italy? America to pay for healthcare and education out of pocket? Italy has plenty to offer a lot of people in this world.

Immigration is quite open in Italy and I personally know a very large amount of people who got citizenship. They are all working elsewhere in the EU, though.