There are three parts to Christ's birth: introduction (Rome's call), climax (birth), and epilogue (the angels' song). The dual purpose, depending on inflection: to accuse and to forgive us of our sin.
Jesus had an odd connection to Rome: a census at His conception and birth (born under the Law), answering appropriately to Caesar in His ministry (render unto Caesar...), and being put to death by Pontius Pilate. We put Him on the donkey to Bethlehem, our sin necessitating His coming. God, five hundred years previously, had told of His orchestration of Christ's census-undergoing.
He was born. Swaddled. Placed in a manger - not a crib - for our sin and for our forgiveness (His passive obedience). A nondescript birth is the climax of this story - how anticlimactic is that? There was no fanfare for Him...yet. Similarly, the Spirit works quietly in our hearts.
Then the angels sang a fanfare - to shepherds, not to Caesar. Why this audience? All God's plan. The glory of God shone, causing inevitable fear (accusation of sin to sinners who cannot abide His glory) as it will on the Last Day, showing all our sin. But the angel said, "Fear not!" Here is the Gospel for everybody! Give up guilt because God's Son has taken it upon Himself (forgiveness of sins). This Savior was born, innocuously, yet bearing all of God's blessings in His wake! Do not reject Him.
Soli Deo gloria.