As to whether or not angels are a part of the authentic Tradition-with-a-big-T of the Church, why, of course they are! When I was in the seminary there was an entire course of theology, called "angelology," devoted to the study of angels.
The teaching of the ancient Church and the testimony of the Bible itself ought to settle the matter for anyone who accepts either of them as authoritative sources. Those of us who accept both as authoritative ought to have even less problem.
Both Old and New Testaments speak often of angels. They appeared, for example, to Abraham before visiting Sodom to try to find ten honest men. Jacob wrestled an angel and had a dream in which he saw angels ascending and descending upon a ladder. There was an angel stationed at the gates of Eden as Adam and Eve were cast out. There was the angel of the Annunciation, Gabriel—without whose presence it is difficult to visualize the Incarnation. There was the angel who appeared to John the Baptist’s father, angels who appeared outside the tomb, and angels who appeared to the faithful at the Ascension—not to mention angels who "ministered" to Christ so often in Scripture. These are far too many references to be simple embellishments, with far too much detail.
True, saccharine images of angels hardly do them justice, and angels like that are certainly not part of Catholic Tradition or theology. Atrocious religious art that depicts them as effeminate creatures of ethereal substance hasn’t really grasped the reality of angels. However well-intended, that kind of thing does no service to the Church or to the faithful.