This is not totally about Italy, but I have seen window screens in Europe.
I stayed in a B&B in Amsterdam that had them. The combo of using them with the fans provided was fine in June of 2018. I was surprised because they were wood-framed, removable and very similar to the ones we used in our 1927 non-air conditioned, non-screened house near Tacoma WA. This is the type I'm talking about. I haven't seen them in Italy, but perhaps someone else has.
I'm another person not particularly bothered by the heat. I grew up in houses in San Antonio without AC. I went to schools without it from 1st through 12th grade. Most of the buildings on the UT Austin campus were air conditioned. But, except for one summer in Austin, none of the places I lived during those 4 years had AC.
We definitely have air conditioning where we live near Tucson. We also have ceiling fans in every room. I like to keep the thermostat at 77 and often wear a light fleece at that temperature. My Tacoma native husband turns it down to 75. I turn it back up. He rarely notices the difference unless he checks the setting.
He and I were in Venice, Rome, Sicily, Sorrento and Genoa mid-September to early October of 2009. I don't remember it being particularly hot anywhere we visited or stayed. Except for the B&B in Venice that only had fans, every lodging was cool enough for us. I'd expect that things would be even better now and in the future.
The one place I've stayed in Italy where I was a bit hot was the 1st hotel (in Padua) on the Village Italy tour in June 2017. The AC in my room simply did not work and there was no fan. The owner tried to fix it to no avail. No one else seemed to have a problem. I chalked it up to my turn in the barrel for "Sleep with ... no (or weak) air conditioning."