This trend is also notable within Greek myth itself. There are three distinct eras: the one of the origins of the Gods, the one in which Gods and Man live amongst one another after Prometheus brings fire to Man, and finally the Heroic Era, in which the Gods are almost absent.

Narratively, it trends towards focusing on Human characters over time above the Gods, so far as we can infer. By the Heroic Era, we are seeing stories almost entirely centered on Human descendants of the Gods such as Heracles or Jason. In contrast to the first era, in which the Gods are the only participants in the stories, they are almost absent except as guiding figures or sources of inspiration. In the middle era, they are described as walking among mortals and actively participating in the world.

There is an interesting throughline in Greek myth, of Mankind being prophesised to supplant the Gods, and that steadily comes to pass over time. The Middle Era deals with Humans who surpass the Gods in places, such as Arachne besting Athena in her weaving, but it is with the Heroic Era that we see Humans who best the Gods themselves, such as Heracles. The Iliad has the Gods making few appearances, and are referenced more in the manner of calling to them for inspiration, courage or aid.

It is therefore likely, that as the Romans took Greek myth and continued it, they also took the mentality of human-centric stories over God-centric ones, and continued in that vein.

For further reference, I would recommend the excellent Mythos trilogy by Stephen Fry, particularly as an audiobook. It's a more narrative overview but it details each era and the interesting continuities and narrative concepts. For specific detail, we can refer to his own source for the stories, Mythology by E. Hamilton, which also discusses Roman, Norse, and other European mythologies.