I may be wrong, but I don't think that's what he's saying there. It seems to me he's talking more about the economic and social impact, although again that sentence is a little florid for my taste.
I'm carefully reading the supplementary materials, since that's usually where you find the real meat of the papers.
I suppose that's indeed what he's saying there. From the Supplementary Materials for A genetic probe into the ancient and medieval history of Southern Europe and West Asia, p. 12.
To the west of Greece, 1 sample from Italy, a Punic sample from Sardinia (MSR002) is identified as Mycenaean-like.(53) We note that the samples from Italy do not include Sicily and Southern Italy at the time or postdating Greek colonization, but they do include a large set of samples from Imperial Rome which we infer to be mostly of Anatolian rather than Aegean or southeastern European origin.
This is more or less what Reich had also said about Anatolia_BA being the source for Imperial Rome
"The demographic significance of Anatolia on a Mediterranean-wide scale is further documented by our finding that following the Roman conquest, the Anatolian population remained stable and became the geographic source for much of the ancestry of Imperial Rome itself. "
https://iias.huji.ac.il/event/david-reich-lecture
From the Science article. There are no samples from Italy in these three studies. So what exactly are we talking about?
"The papers also acknowledge the nuances of identity in later periods, for example in Imperial Rome. Previous genetic studies had shown that as the empire coalesced, the ancestry of people in and around the city of Rome shifted, with most having roots not in Europe, but farther east.
After obtaining dozens of additional Roman-era genomes from the region, the team zeroed in on the source of those newcomers: Anatolia. But the researchers agree that people with “Anatolian” DNA moving to the Italian peninsula likely saw themselves as citizens or slaves of Rome, rather than as part of a distinct “Anatolian” ethnic group. Contemporary chroniclers remarked on the new faces in Rome—and referred to many of them as “Greeks,” perhaps because the eastern peoples had spoken Greek for centuries, Lazaridis says."
https://www.science.org/content/art...provides-clues-origin-farming-early-languages
First criticism from archaeologists
"Some archaeologists still think the papers claim too much influence for ancestry. “DNA cannot tell us anything about how people shape their life worlds, what their social status was,” says archaeologist Joseph Maran of Heidelberg University. He says terms like “Yamnaya ancestry” suggest the Yamnaya spread by moving directly from place to place, rather than through a complex mingling of their descendants with local populations over centuries or more. “Equating history with ‘mobility’ and ‘migrations’ is … old-fashioned.”