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The New Covenant is the sacrifice of Christ for the forgiveness of sin verses the Old Covenant of living by the law of Moses. The Old and New Covenant is compared in Hebrews Chap 8-9 and was foretold throughout the Old Testament (for example Jeremiah 31:31-33) and explained in the New Testament.

If people want to take Luke 22:20 literally

And in the same way He took the cup after they had eaten, saying, “This cup, which is poured out for you, is the new covenant in My blood.

They would have to say it was the cup that died for our sins.

If people want to take John 6:52-54 litereally,

Then the Jews began to argue with one another, saying, “How can this man give us His flesh to eat?” So Jesus said to them, “Truly, truly, I say to you, unless you eat the flesh of the Son of Man and drink His blood, you have no life in yourselves. The one who eats My flesh and drinks My blood has eternal life, and I will raise him up on the last day.

Then they would have to conclude that Christ was literally referring to cannibalism and their response would be offensive (I hope).

There is nothing mystical about communion. Christ was simply making a point that we must repent to God and accept His death that He paid for the forgiveness of our sins. That is the new covenant. There will not be another covenant. Christ left this sacrament (communion) so that we could proclaim what He has done for us. Communion physically ties us to a direct act of Christ and a way in which we can proclaim Christ. That is a great honor and blessing in itself.