Ambrose of Milan

But a saint is not only the blessing of his parents, but also the salvation of many; as it follows, And many shall rejoice at his birth. Parents are reminded here to rejoice at the birth of saints, and to give thanks. For it is no slight gift of God to vouchsafe unto us children, to be the transmitters of our race, to be the heirs of succession.

Catena Aurea by Aquinas ↗

Bede

And there will be joy to you and exultation, and many will rejoice at his birth. The father rightfully rejoices because either born in old age or has received a son of such grace. Others also rejoice, to whom he will preach the hitherto unheard entrance of the kingdom of heaven. And it should be noted that many rejoice at the birth of the forerunner, but at the birth of the Lord, the angel announces great joy which will be for all the people. …

On the Gospel of Luke ↗

Bonaventure

It is described lastly as consoling in the words: And you shall have joy and exultation: Proverbs 23: "The father of the just exults with joy," that is, Zacharias, the father of John; and "he who has begotten a wise son shall rejoice in him," because Proverbs 10: "A wise son makes the father glad." — Nor shall that joy be singular, but common. …

Commentary on Luke, Chapter 1 ↗