One more time: the catechism says the eucharist is instituted not in John 6, but at the last supper, and Scripture says the new covenant is instituted at the death of the Lord.
The Eucharist is instituted at the last supper. Jesus said, ‘Do
this in remembrance of me.’ What is ‘this’ that Jesus is talking about that we are to do in remembrance of him? ‘this’ is the Eucharist where Jesus takes bread and says, ‘this is my body’. And he takes wine and says, ‘this is my blood.’
In John 6 Jesus is making a covenant with those who would believe in him. He commands those who believe in him to eat his body and drink his blood in order to have eternal life. That is a command with a promise that Jesus will give us his body and blood to eat and also give us eternal life if we eat it. It is a covenant between him and those who obey his command.
This covenant was first manifested at the last supper when Jesus provided his body and blood for the disciples and continues to be made manifest at each Eucharist by his commandment to his disciples to do this in remembrance of me. Jesus said, ‘This cup
is the new covenant in my blood; do this, whenever you drink it, in remembrance of me.’ (1 Cor 11:25). Right there in scripture it says ‘this cup is the new covenant.’
The sacrifice on the cross is Christ making good on his promise. As he said, in Jn 6:53, ‘the bread that I will give for the life of the world is my flesh.’ Jesus is a priest in the order of Melchizedek because he offered a sacrifice of ‘bread’ just like Melchizedek. But Jesus’ bread is no ordinary bread but it is his own flesh offered as a sacrifice on the cross made manifest to us through the Eucharist.
Without the Last Supper no one would know that Jesus’ crucifixion was a sacrifice. What Jesus instituted at the Last Supper was a covenant that was ratified on that cross. As you said the will and testament is put in effect when the person dies. But, the will and testament is written and agreed to
before the person dies. John 6 and the Last Supper gives us that will and testament of Jesus. Thus, it is entirely feasible to say that the new covenant was instituted at the Last Supper and put into effect through Christ’s death and Resurrection.