No, it doesn't. You just stop listening when you hear the answer you want... or don't listen to an answer you don't want. Either way.
In John 6:63, Christ is speaking of our fleshly limitations. We can't understand His teaching because we see with carnal eyes where He is speaking of Spiritual matters. "The" flesh (not "My Flesh") is a New Testament term often used to describe our human nature apart from God's Grace. Here, Our Lord is reiterating that no once can come to Him (understand Him) unless drawn by the Father. Those of the flesh do not understand matters of the Spirit. That is His explanation for why so many left Him. On another level very closely related to that last point, Christ said, “It is the spirit that gives life, the flesh is of no avail,” because He wills to eliminate any possibility of a sort of crass literalism that would reduce His Words to a cannibalistic understanding. It is a cross-understanding with the verses before which speak of His Ascension. The Ascension and the distribution of His Body and Blood in the species of bread and wine are works of the Holy Spirit--as is our belief in the testimony of Christ.
At the Last Supper, Our Lord didn't raise the flesh of the lamb and proclaim it to be Him--though He is the Lamb of God. He raised the unleavened bread. It isn't the form of the flesh that matters but the Spirit which gives life. This is the Transubstantiation that afflicts so many on the outside and yet gives life to the Church. What you think is a modern "gotcha" for our way of thinking is actually a matter of deep theological insight for the greats among the Doctors of the Church.
That makes no sense. The focus of the conversation was the "eating of the flesh" and the angst it caused the listeners. There is no change in direction of the conversation.