In THE AENEID there is much made of Rome being founded by descendants of Aeneas (and Teucer and Dardanus and Ascanius) and there are many mentions of Rome as a "second Troy" and descriptions of reprisals against Greek city-states (like Lucius Mummius' sack of Corinth in 146 BC), for the destruction of Troy.
Forgetting for a second the historical veracity of the Trojan War and Teucer/Dardanus/Aeneas/Ascanius, can someone explain how seriously Romans thought of themselves, even spiritually or figuratively, as descendants of Troy?
Did they feel like hardy wounded survivors who had had a second chance or rebirth? Did their "nostros" feel like a kind of Trojan manifest destiny?
Or was this story about a supposed Trojan heritage just a nifty part of an epic poem written to please Augustus? Was it not really taken seriously by the Roman people?
This is a very, very good question. Sadly, given the sources I have, there is no way I can really tell you what the "Roman people," thought about anything. I can only tell you what's in the surviving texts.
Given the texts we have (and the fact that I can't tell you the intentions of the ancient authors with certainty), I always assumed that the whole "descent from Aeneas" thing was used mainly for literary and propagandistic effect. The proem of De Rerum Natura is a very convenient example of the literary use. Lucretius, being Lucretius, obviously doesn't believe that Venus is really the mother of Rome via Aeneas, nor does he think that she can stop the civil wars by satisfying Mars. But when he writes about the "Venus Genetrix," he's really paying homage to epic conventions (recalling the invocation of the muses which was customary for an epic proem) while making a salient political point (civil war bad, want peace please). DRN was almost immediately popular with the brothers Cicero and Caesar himself (who many Classicists believe to have been a Epicurean like Lucretius), so I am inclined to say that other elites also saw the story in the same light.
This is a nice place for me to segue into the propagandistic use of the story. Despite not being religious, Caesar was more than happy to claim descent from Venus. Despite being ancient and Patrician, the Julii were only technically noble (I believe there were only 3 consuls in his family). Being the direct descendant of Venus Genetrix improved his pedigree. Thus he put her on coins, in his basilica, and built her her own temple. You could argue that Caesar's promotion of the story suggests that the average Roman was culturally invested in it, but I don't think we have enough evidence to support that conclusion.
This is a nice segue to the final bit. Forgive me also if I'm misunderstanding the question, but when you ask if the Roman people "took the story seriously," I'm assuming you mean, "did the average Roman think the story was literally true?" This is a tough question because, very generally speaking, Roman historians didn't give a single fuck about whether something was literally true or not. In their minds (again very generally speaking), the goal of writing a history was didactic value. They repeated stories because they believed these stories taught moral lessons or otherwise had some value to the reader. It doesn't seem like they considered this dishonest in any way, they just had a different set of goals. So Caesar probably wouldn't have thought there was any problem with being an Epicurean on the one hand and claiming descent from Venus on the other. The story had some moral value as you very rightly point out and it had political value for him. To him and almost every other Roman intellectual, that would have made it good history, regardless of whether or not it was actually true.