When in the old tradition of the Mass the Celebrant chanted the Our Father alone, he was not only following a venerable tradition which dates to the Early Church--which ought to be reason enough to retain this noble practice--he was also serving as bishop, the symbol of the unity of the Church. It is the same as when he chants the Collect alone (which, by the way, he should also do alone, especially in the Mass). It is called the collect because it is the prayer of the collected faithful: the many become one, expressed by the one mouth of the bishop.
Similarly, the Our Father is a prayer in which the many become one. The celebrant speaks for all the people, he performs a priestly duty here. But more importantly it is a Christic duty; in performing this duty he is serving as Christ for the people, presenting their petitions to God out of his own mouth.
After all, what gives us the right to call God our Father? Is He not the Father of Jesus Christ? And is not Jesus the only-begotten Son of God? God has no other children than Christ: he is the only. Then how is it that Jesus bids us to call God our Father? Surely, this can only be properly understood as an invitation for us to pray in Christ. We pray this prayer to God as though we were Christ Himself, for we are in Him. Heathen cannot rightly pray this prayer. It may not be prayed by one not baptized. This is why the pastor lays his hands on a child presented for Baptism, during the Our Father: it is an indication that the Our Father is here, in Baptism, being given to this candidate as a gift: he is being incorporated into Christ. Now, having been baptized into Christ, he is privileged to call God Abba, Father.
When at the High Feast of Salvation this principle is most properly expressed by the celebrant's utterance of this prayer alone. All Christians say this prayer day by day; but at the Altar, where the many become one, so the prayer fittingly becomes uttered by one voice, the voice of Christ. Christ employs the mouth of His servant, the celebrant, but the voice is most certainly His own. And thus all the people pray this prayer as one.
And it is said in immediate proximity to the Verba, the words of Institution, spoken also by the celebrant alone, in the stead of Christ.
Adapted from a 1999 article. Father Eckardt will be on vacation until mid-August. Comments will not be answered until then.