The term "Body of Christ" can be taken several ways. It can be taken to mean the mystical body of Christ, into which all believers are received when they are baptized. St. Paul thought of the Church in that way, expanding the analogy to different members with different spiritual gifts being different parts of the body--the eye, the hand, the foot, etc. I don't think, however, that that is the meaning assumed by the solemnity that we recently celebrated.
Consider the readings. In Cycle A, the gospel is from the Bread of Life discourse, John 6:51-58. In Cycle B, the gospel is from Mark's institution narrative, Mark 14:12-16,22-26. In Cycle C, the gospel is from Luke recounting of the multiplication of the loaves, Luke 9:11-17. Add in the other New Testament readings (1 Cor 10:16-17, Heb 9:11-15, and 1 Cor 11:23-26), and you have an even stronger emphasis on the institution of the Eucharist at the Last Supper, when Christ took bread, blessed and broke it, and said, "This is my body."
Throw in the sequence before the gospel, and it is clear that the Church intends this solemnity to be a celebration of Christ's real presence in the Blessed Sacrament. Focusing on the members of the Church as the body of Christ is not wrong, but I think that it misses the main point and risks undermining it.
Similarly, we do well to remember that even in a poorly performed (even illicit!) liturgy, the miracle of transubstantiation takes place, leaving no longer bread and wine, but the Body and Blood of Christ.
Fatima Prayer to the
Oh Most Holy Trinity,
Father, Son, and Holy Spirit,
I adore Thee profoundly.
I offer Thee the most precious Body, Blood, Soul and Divinity
of Jesus Christ, present in all the tabernacles of the world,
in reparation for the outrages, sacrileges and
indifferences by which He is offended.
By the infinite merits of the Sacred Heart of Jesus
and the Immaculate Heart of Mary,
I beg the conversion of poor sinners.