The Additional Directions in the 1979 Book of Common Prayer contain these instructions.
“During the Great Thanksgiving it is appropriate that there be only one chalice on the Altar, and if need be, a flagon of wine from which additional chalices may be filled after the Breaking of the Bread.” p.407
Why just one chalice? There are several reasons.
First it best depicts the historical narrative. The text says “In the same way he also took the cup, after supper saying….” Jesus took the cup. He did not use many cups nor did He distribute individual cups to the disciples.
Second Jesus said, “This cup is the New Covenant in my blood.”
This one cup signifies the New Covenant that Jesus makes with His Church. He did not make an individual covenant with each disciple rather He invited them and us into His one New Covenant.
Third this one cup signifies that we are one Body. A Roman missal puts it well. “The presence on the altar of a single chalice and one large paten can signify the one bread and one chalice by which we are gathered into one Body by the Holy Spirit and may truly become a living sacrifice in Christ.”
American Christianity is fractured by individualism so perfectly represented by the egregious practice of distributing the Blood of Christ in tiny plastic cups. This is a celebration of “me and Jesus” rather than St. Paul’s teaching in 1 Corinthians of the Church being one Body. There is a lovely line in in one of our hymns.
As grain once scattered on the hillsides
was in this broken bread made one
so from all lands your church be gathered
into your kingdom by your Son.