Upvote:2
The priest celebrates Mass In persona Christi (in the person of Christ). As Pope Pius XII explains,
The priest is the same, Jesus Christ, whose sacred Person His minister represents.
Given this liturgical principle and reality, it is appropriate that Christ's action (washing of feet) is performed by the priest instead of the laity.
The consecration of bread and wine into the Body and Blood of Our Lord is essential to the Eucharistic sacrifice; washing of feet is not.
Psalm 110 tells us that Christ will be "a priest forever in the manner of Melchizedek." The essence of Melchizedek's priesthood was offering "bread and wine".
The Mass is a true sacrifice, not merely a commemoration of the Last Supper. What is offered at Mass is Christ's Body and Blood; what is consummated at Mass is Christ's Body and Blood. Washing of feet is neither offered nor consumed.