The practice is so common that many people might be under the impression that this hymn is integral to the Roman Rite. In fact, it is not. To use this hymn as the entrance is a substitute - the last option permitted in the rubrics, presumably only to be used when no other choice is possible - for the authentic Introit, which is Ecce Advenit:
The text translates as "Behold the Lord the Ruler is come: and the Kingdom is in His Hand, and power, and dominion. Give to the king Thy judgment, O God: and to the king’s Son Thy justice."
The text itself is extremely important for understanding the core of the drama concerning the life of Christ on earth. The perception that he had come as an earthly ruler was shared widely and was the basis of his following and also his execution. Time and again he is greeted as a King - here in this scene and also on his return to Jerusalem that we celebrate on Palm Sunday. The fullness of the truth of his life and death would be revealed only much later: that his kingdom was not of this world but of eternity.
The text, then, is important, and is not captured in "We Three Kings" - which was not written for Mass at all but rather for a popular folk pageant that tells the Christmas story. It is not a piece for liturgy.
We should also look at the Gregorian melody to see that it is the first of many of the propers this day that have an intriguing hint of Eastern style. It is in a minor mode and doesn't move very far up and down the scale, and the same can be said of other propers this week. There is a certain "foreignness" to them that not only suggests all things East but also draws our attention to the universality of Christ, on both heaven and earth. The ear of the Roman Rite Christian, these chants do act as a signal.
And yet, even if we dispense with the melody, we do better to sing the proper text than to sing a generalized hymn at the very start of Mass. The entrance proper help us with the critical task of orienting ourselves in a liturgical manner. Our purpose is not merely to gather, sing, and worship. Instead, we are looking forward and processing together toward a purpose and a with a direction in mind.
There is no better time to begin this orienting process than at the very start of Mass. Replacing the proper with a hymn delays this orienting posture until later - and thereby subtly inhibits us from coming together in directed prayer.
There are other options for singing the propers on Epiphany. The Anglican Use Gradual offers this:
The Complete English Propers by Abrogast offers:
The American Gradual offers another in English that also preserves the Gregorian melody.
Our own choice is for the choral proper by Richard Rice from his own book.
Whatever setting of the proper chosen on this day, it is a more fitting way to begin Mass than with a people's hymn that is designed for popular and not liturgical occasions.