The Pharisees were offended that Christ would eat with sinners (Luke 15:1). They didn't know the half of it: He came not merely to eat with sinners, but to give His flesh to be their food and His blood to be their drink (John 6:48–58).
It can be amusing to hear people speak about the latter part of John 6. I agree that sacramentalism has done its best to ruin this amazing chapter, but we oughtn't let fear of sacramentalism keep us from entering into what the Holy Spirit teaches here. It's obvious that Christ isn't literally speaking about eating His flesh, for the simple fact that He also taught His own resurrection. You have a bookkeeping problem if you try to believe in both Resurrection and literal eating of the flesh of Christ.
That being said, the Lord chose deliberately provocative language to describe His giving us life. We've noticed before that the Son of God can give life merely by calling the dead from the tomb (John 5:25). But when the Son of Man gives life, it costs His flesh and His blood. But I don't think that's all there is to John 6. There is not only His giving, but our eating and drinking. It's not just that we believe on Him (we do), but we must feed on Him as well.
1 Corinthians 10:15–23 brings this into the context of the Lord's table. 1 Corinthians 10:15–18 makes the association between our eating the loaf and drinking the cup and the altar. It takes us back to the Numbers 18:8–19, there the priests were to eat all the heave-offerings the people presented. 1 Corinthians 10:16 tells us this means they had communion with the altar.
There is an association between the Lord's table and the altar. We are making a statement about that association every time we break the one loaf and drink from the cup. We are claiming our communion with the death of Christ. By eating the one loaf and drinking from the cup, we are saying we are in fellowship with the sacrifice.
I don't question that we are to feed on Christ individually, but the feeding in 1 Corinthians 10 is corporate: we being many, are one Body (1 Corinthians 10:17).
The Old Testament sacrifices were all assumed to be more than enough: with a couple exceptions (Leviticus 6:23; Leviticus 6:30), there was something for the priest in every sacrifice. Even the burnt offering, which was wholly consumed, had a part for the priest – the priest gets the skin (Leviticus 7:8).
Of course they weren't really more than enough, but the principle was established. Really, the blood of bulls and of goats is incapable of taking away sins (Hebrews 10:4). But the sacrifice they pointed to – Christ offering Himself for us by the eternal Spirit (Hebrews 9:14) – that sacrifice was far more than enough.
Christ was both our sacrifice and priest. We, as family of the priest (Numbers 18:19), are to eat of the sacrifice. By feeding on the sacrifice, we express communion with the altar (1 Corinthians 10:16). What does it mean to have communion with the altar? At the very least, it means we recognize and agree with the need for the sacrifice. At the very least, when we contemplate feeding on Christ, we contemplate our deep need of Him.
I don't doubt there is the feeding on Christ in Resurrection as well as feeding on Him in humility. It seems to me John 6 is talking about the former: it is the One who has come down from Heaven.