I am not a cultural historian, but I understand that there might be a (tenuous) basis in fact for the idea that Rome is connected to the inhabitants of Troy.
There is some evidence that the Etruscans actually were descended (in part) from a group of people who migrated from western Anatolia. The similarities in the written languages, particularly the small connecting words, of Etruscan and Luwian are apparently fairly significant. I don't have the book in front of me, but I saw several side-by-side Etruscan/Luwian word lists that were remarkably similar (by 'similar', I mean that over half the words were identical, and most of the rest had minor but regular pronunciation changes). Plus the
Lemnos stele indicates a connection between the two.
It seems entirely plausible to me that a group of Luwian-speakers may have migrated in the late Bronze or very early Iron Age from the Troy area of western Anatolia to Etruria, forming the core of the future Etruscan culture, and that legends of this event were passed through the Etruscans to the Latin-speaking Romans (who were culturally dominated by the Etruscans for centuries). Livy would presumably have taken these legends and shaped (parts of) them into the Rome-centric version we know of today.