Theophylact of Ohrid

Before God, only the pure serve as priests, while from the unclean God turns His face away.

Commentary on Luke ↗

Augustine of Hippo

The priest was offering sacrifice on behalf of the people. The people were expecting the Christ. John was the one who would announce the Christ.

Sermon 291.3 ↗

Bede

Now it happened that while he was performing his priestly service before God in the order of his division, according to the custom of the priesthood. Indeed, through Moses the Lord appointed one high priest, who, after his death, commanded one to succeed in turn, and this was observed until the times of David, from whom it was decreed that there should be several (as we have said), nevertheless by the Lord's action, who, ministering in turns, during the time of their division, each would dedicate themselves to chastity, and would not …

On the Gospel of Luke ↗

Bonaventure

Therefore, indicating the due execution according to the congruence of order, he says: It came to pass, while he executed the priesthood, etc.; whence that which is said of Aaron in Ecclesiasticus forty-five applies to him: "The Lord gave him to exercise the priesthood and to have praise, and to glorify the people in his name," etc. …

Commentary on Luke, Chapter 1 ↗