The question is whether or not their 'Italian-like' admixture is from actual Italians or not. As clearly Ioanian islands as a whole do not score totally like mainland Greeks on any commercial site or admixture test. Maybe Corfu could be an exception here, but all the rest show a pull towards Italy and West Asia compared to the mainland.
About this topic though, I don't think the Y-DNA results are particularly enlightening unless the genealogy of each person is well filtered to make sure only people who's paternal line has a long presence are included in the statistics.
I can tell you now that E-V13 is absent in Kalymnos apart from a small number of recent migrants. As for other Dodecanese islands, I cannot safely say. But I will say that I have a few E-V13 matches from Symi and their paternal lines are confirmed to be native there for many generations. And them scoring almost 0% Greek & Balkan is a good indication it's not due to a recent mainland ancestor.
As for which clades Proto-Greeks carried, I don't think studying the modern population of Greece is particularly useful for this subject. Ancient DNA is the most important. Although if the modern route is insisted, perhaps Northern Tsakonians who were quite differentiated even from Maniots would be much more representative than the average modern Greek.
Worth of investigation possibly is subclade I-S12195. It was found in one Yamnaya individual and also exists in quite a few of my relative matches from Greece, even those outside of Kalymnos. According to Soreclow's stats, it also had a high frequency in certain a certain area of the Peloponnese. Although it seems relatively quite rare overall so it probably was a minor clade anyways. So perhaps not worthy of investigation after all.
