Part of it was that myth was a big part of Roman identity. The book "the Birth of Classical Europe" by Simon Price and Peter Thonemann talks a lot about how myths, among Greek city states then their Roman successors, began to be used as links between peoples or as justifications for certain alliances or political actions. Cities would find common mythological ancestors or mythological ancestors to justify alliances. This went back centuries before Rome was important and it meant that myth became a natural part of Roman politics. By tracing a founding ancestor to one of the Greek heroes of the Illiad, the Romans made themselves as ancient and illustrious as the Greek city states and justified their alliance and rule over the Greek lands - since they were founded by a Greek, they weren't random foreign conquerors at all!

In the case of Julius Caesar and then his successor and adopted son Augustus, the link to Aeneas was a way of justifying their leadership. If you check out the Aeneid by Virgil - which became so popular and influential that it's one of the canonical texts of the Western world even today - it was commissioned by a political ally of Augustus. (I got this from the translator's postscript in Robert Fitzgerald's translation of the Aeneid.) So obviously there was some political motivation there since it portrays Rome rallying behind Aeneas - the reader was supposed to draw a parallel between the ancestor and the descendant. So it was supposed to convince people to follow Augustus.

Roman political culture in general was based on some of these myths, since the communal identity of Rome and a big unifying element was their foundation myth including heroes and their feats, as well as the rituals that were given to the Gods in order to sustain the Pax Deorum, or the favour of the Gods. The idea of Rome as chosen by the Gods was an influential one to the Roman population; but it shouldn't necessarily be viewed as pure propaganda since it seems to have been a genuine belief of many Romans, including leaders.

Personal claims to ancestry like these seemed to have become more common in the late Republic. For example, all three of the Triumvirs in the first Triumvirate claimed different mythological ancestors, according to the paper "Descendants of Gods" by O. Hekster. It was mostly prominent Patrician houses that had these claims. So, not having these claims would have been most common among the "new men", or the men who became the first senators in their families. These would have been wealthy men, able to pay the 800,000 sesterces needed to join the Senatorial class and make a start in the "Cursus Honorum". They would have been seen a bit the same way as the nouveau riche are seen by old money today - you can find accounts of them being looked down on. For example, there was a Roman novel called the Satiricon where "new men" are made fun of for being gauche and awkward in high society. Their lack of heroic mythological lineage would have been perceived as part of being the "new man" making his way into the elite circle, so it was added onto the more general stigma among senators of being a "new man". However, the practice of claiming mythological descent seemed to have died out after the Republic, as the emperor began to try and have sole claim to such amazing ancestry (Hekster, 2-5).

However, I think the distinction between "real relations" and "mythical people" you make is one that wouldn't be so clear to the ancients themselves. If you look at Greco-Roman histories, like Herodotus, they seem strange indeed since there are monsters and gods and heroes throughout it. So this wasn't just propaganda, necessarily; it was a part of their authentic view of history. (Which is part of what made it so effective when someone used it to their own advantage - other politicians and regimes have used history in this way in other cultures too.)

(This isn't my number one area of expertise so I'd love some other contributions.)