Luke 2:11 says,
Luke 2:11 For unto you is born this day in the city of David a Savior, who is Christ the Lord.
The verse declares Jesus to be the Savior, but at that point, he hadn't yet saved us, for he had yet to suffer for our sins.
Isaiah 53:5 But he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed.
1 Peter 2:24 He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed.
Additionally, to be the Christ means to be the anointed. The only time the Bible declares that Jesus was specifically anointed was at his baptism.
37 You know what has happened throughout the province of Judea, beginning in Galilee after the baptism that John preached— 38 how God anointed Jesus of Nazareth with the Holy Spirit and power, and how he went around doing good and healing all who were under the power of the devil, because God was with him. - Acts 10:36-37
21 Now when all the people were baptized, and when Jesus also had been baptized and was praying, the heavens were opened, 22 and the Holy Spirit descended on him in bodily form, like a dove; and a voice came from heaven, “You are my beloved Son; with you I am well pleased.” Luke 3:21-22
It was only after Jesus's baptism does he declare
18 “The Spirit of the Lord is upon me,
because he has anointed me
to proclaim good news to the poor -Luke 4:18
Since Jesus had not saved us by merely being born and since he was not anointed until after his baptism, should we thus understand Luke 2:11's identification of the child as "a Savior, who is Christ the Lord" to be a prophetic identification or designation - a role Jesus was to fulfill and not his actual status since he had not yet saved or had been anointed?