Sigh.
You don't know what you're talking about, like at all.
First of all, lets get to the DNA side. Not even all Italians overlap with each other. Take the Sardinians for example, they're far away genetically from all other Europeans including Italians despite being located in Italy. Basque are distant from all other Europeans as well geneticaly, yet still look European.
A little history lesson for you. The concept of "whiteness", came from the Spaniards, yes, not Nothern Europeans. The Brits adopted the concept of whiteness and the caste system from the colonial Spaniards.
The Irish were not accepted in the Klan either, and just like the Italians it was because they were catholic. NOT because of their ethnicity. Both the Irish and Italians had the SAME rights every other white did in the USA. There was never any point where the Irish/Italians went through the same issues that African Americans did. They experienced lite racism and xenophobia just like every other non-english immigrant did back then. But lets look at the facts. 1) Both Irish and Italian Americans were never segregated, they went to the same schools all other whites did. 2) They drunk out the same fountains other whites did. 3) They were always allowed to inter-marry with other whites. So stop this bullshit popular narrative, that either of the groups were at any point considered non-white. You can dig up old Italian-American documents from as late as the 30s and see that they were listed as Caucasian. Hell a handful of Italian-American actors and singers were active during that time period and all were considered white.
Back to the DNA bit, ALL European ethnic groups are distant to others slightly. The Russians are distant from the British, does that make Russians non-whites? The British are distant to Poles, are either of these non white? The Germans are distant to Lithuanians. We can go on and on, but all of these groups are closer together than they are any other groups on Earth due to all of them originating from a common people who migrated out of the Caucasus. Italians, Spaniards, Greeks etc share nothing in common with any continent but Europe. In fact the only people who you can consider "fringe european" who comes close to only Southern Italians, are Ashkenazi Jews, and that is because most of Ashkenazi Jewish mtDNA comes from Southern Italians. Ashkenazi Jews are in essence partial Southern Italians maternally.