the day of the election ... and I am able to proceed.
It could have been made more clear to your readers if you had indicated that "the election" you were talking about were local elections - presumably means that it is not about the election of national leaders.
I think in most US locales local residence is all that's required for that as well.
The national IDs being discussed are in the context of a federal election - the presidential election of 2016 for example. It would be interesting to hear your experience, as an Italian citizen, to vote for the presidents of the Netherlands or Germany.
In the US, a lack of uniform national ID, and some states' laws prohibiting ID checks and others states issuing state IDs (without citizenship proof) have made voter fraud possible, and likely. the latest example came out of a political science researcher Old Dominion suggested that Hillary may have received as many as 800K votes from illegal aliens -> as many as 6%+ of non-citizens voted in the last presidential election.
and if you go through his earlier papers on non-citizen voting, you will find a large body of literature / researches in this area and it is fairly conclusive that non-citizens do vote in national elections consistently.
I think so far the trump administration has not provided much basis to say that this is a widespread problem. on the other hand, his political opponents, including the media, are equally baseless to call trump's claim "false" - as they don't have any facts to dispute his claim either.
With that, it makes sense to get to the bottoms of it, and see just how big / small a problem illegal votes (including non-citizen votes) are.