You are right, but only as far as you go. He also said:
"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 you; he who eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day. For my flesh is food indeed, and my blood is drink indeed. He who eats my flesh and drinks my blood abides in me, and I in him. As the living Father sent me, and I live because of the Father, so he who eats me will live because of me. This is the bread which came down from heaven, not such as the fathers ate and died; he who eats this bread will live for ever." (John 6:54-59)
What is different at Mass is the mode or manner of offering. It is to this sense that the Catholic refers when he says that the Mass is un-bloody. But in the Mass, bread and wine transubstantiate into the Body and Blood of Christ. So obviously, in that sense, the Mass is a bloody Sacrifice. It does (and must) contain the true Body and Blood of Christ, otherwise it could hardly be identical to the Sacrifice on Calvary. However, whereas on Calvary, Christ died and shed His Blood in a unique way, in the Mass our Lord mystically renews His death and Body-and-Blood Sacrifice in a sacramental way, not under the appearance of His Body and Blood, as on the Cross, but under the appearance of bread and wine; hence the manner of offering at the Mass is unbloody. It is bloody in the sense that it is the Body and Blood of Christ, but unbloody in the sense that it is offered under the appearance of bread and wine in a sacramental fashion.
Hope that helps.
God bless!