Lydia was actually on the west coast of Anatolia, not the south coast...not too far from the area of Troy. The Romans believed they were descended from the Trojan Aeneas, who had survived the fall of Troy and made his way to Italy, but the Etruscans seem to have had an interest in Aeneas as well--probably earlier. There are at least 17 vases from the 6th and 5th centuries BC showing Aeneas, 10 from the one Etruscan city of Vulci.
Perhaps there is some connection between the Trojan refugee legends and the Lydian legends. Herodotus claims to be giving the Etruscans' own version of their past, and a number of other Greek historians also assumed it to be true.
On the other side of Greek historians was Dionysius of Halicarnassus, who was the first to suggest that the Etruscans were indigenous...mostly because (and not a bad reason either), they didn’t agree in language or customs with any people they were said to be related to in Anatolia.
Massimo Pallottino, the famed Etruscologist, has pointed out that whatever the origins of the Etruscan people, their civilization as we know it developed in Italy, and there I think he’s quite right.
Good point about the Etruscan vases with the Aeneas legend...it’s definitely a VERY old story in Italy.