Clearly, once again you didn't read the link. That was one of the founding myths, just like it was a founding myth in Rome. How the heck do you know it wasn't believed in the Britain of that time? Are you another mind reader, only this time able to go back a thousand years?
You are also mistaken about the Greek input. I think it was large enough in Massalia to have an impact, probably in Liguria too, or where did all that E-V13 we carry come from?
Same for central Italy around the Adriatic. That's a big hotspot in Italy for J2 and there is abundant archaeological evidence for contacts with Greece. You know, the big green blob NORTH of Rome in the map you posted.
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As for the Phoenicians, I do think they might have had a small impact in northwestern Sicily and Sardinia, the only places where they actually had their own settlements. It just didn't extend beyond that. Also, the Greeks went in for folk migrations, and imo the Phoenicians, at least originally, were much more about just establishing trading emporia and exploiting natural resources. Sort of like the European cantonments in 19th century China.
It's amazing to me how people distort my positions.