Not everyone who partakes in your Eucharist or attends another denomination is a Christian...THEREFORE, partaking in the Eucharist doesn NOT link the Body of Christ...If it did, there's be a lot of non Christians in the Body of Christ...
Only Christians make up the Body of Christ...
Simply partaking of the Eucharist without faith or belief in what one is receiving does not make one a member of the Body of Christ. Is that your point? If so, I agree. As usual however, you've got half the story. Because some who receive the Eucharist do so without faith in Christ, does not in any way detract from what the Eucharist is intended to be by Jesus nor does it alter in any way the importance of the Eucharist and what it does for those who believe.
For those who believe, the Eucharist is indeed the bond which unites the Body of Christ. It's not a question of whether faith or the Eucharist unite us to the Body of Christ. Both faith and the Eucharist are necessary. The Eucharist is indeed the Body of Christ who is the Head of the Church, as St. Paul tells us. How could the Eucharist be anything but the spiritual food which unites us to God if it indeed Christ himself whom we are receiving?
That is why Jesus said:
Amen, amen I say unto you: Except you eat the flesh of the Son of man, and drink his blood, you shall not have life in you. 55 He that eateth my flesh, and drinketh my blood, hath everlasting life: and I will raise him up in the last day.
John 6: 54-55
John's Gospel treats of this at some length. Furthermore, the doctrine of the Mystical Body of Christ as expounded by St. Paul makes it clear that we are the members and Christ is the Head. If Christ is a) truly present in the Eucharist and is b) the Head of the Body, surely partaking of that Eucharist with faith binds one to Christ?
Jesus' final act with his apostles before his Passion and death was the First Eucharist, the Last Supper. This tells us that although Jesus was no longer to be physically present with the apostles they would be united with Him and with each other in the Eucharist.
Only Christians make up the Body of Christ...
True.
That begs the question of who or what is a Christian? See above.