This probably won’t be very helpful, but here goes.
The only Catholic reference to the Sybils I have ever seen (other than historical stuff) is in the Latin version of the “Dies Irae”, one of the old hymns at Catholic funerals. If I can remember it properly.
Dies Irae (day of wrath)
Dies Ila (day of mourning)
Solvet Saeclum (when the world is reduced)
In favila (to ashes)
Testae David (foretold by David)
Cum Sybila (and the Sybil)
It was kind of terrifying; about the end of the world and our judgment. It was, however, both preceded by the comforting “Requiem” (rest) and followed by the “Voca me” (hear me). Mozart wrote a Requiem Mass, and it’s beautiful. There are others, but the words are always the same.
Anyway, my belief is based on my experiences with medieval and renaissance literature. Literate people back then were enormously educated in the classics; far more than now. Chaucer himself was a wool merchant by trade, yet his mastery of the classics was vastly greater than that of most classics professors today. (We’re not half as smart nowadays as we think we are.)
Anyway, they utilized many, many classic references in all kinds of ways, including in purely Christian sacred works. They did that, not to affirm, e.g., whether there were Sybils (there were) or that they somehow were truly prophetic. (Though there were some prophecies of the Cumaean Sybil that seemed to foretell Christ, and the Church has always wondered about that.) but as a sort of shorthand way of expressing a lot more.
So, in the Dies Irae, for example, the point was not to say some Sybil or other was truly prophetic, but more to say that certain kinds of knowledge of sacred things were accessible to human beings even without Revelation. It’s like an affirmation of human reason’s ability to know something of God and His will and plans.
So, if even the pagan Sybil could figure there would someday be an end, and that we would be judged by whatever Divinity the Sybil believed in, how much more should we, who have Revelation, know? The reference to both David (a true prophet of God) and the Sybil (natural reason) unite faith and reason in the knowledge that we shall be judged for our lives.
Clever folk, those ancestors of ours. They said lots of things like that.
A more recent example of the same thing is T.S. Eliot’s “The Waste Land” in which the explanations of his references to other works are massively bigger than the poem itself. Again, it’s a “shorthand” way of expressing a lot in a compressed sort of way.
That’s my best explanation.