Being a former Limited Atonementist and believer of other Calvinisms, I understand sky's eagerness to combat the theology of Calvinism but in this case I don't believe his construct is accurate, but more importantly it is not necessary in John 6.
Most often it is the interpretation of verses 35-50 that Calvinists believe lend weight to their doctrine, particularly that of election. However, the passage itself need not be strained in order to challenge the Calvinist view.
Specifically, our Lord's statement:
37All that the Father gives me will come to me, and whoever comes to me I will never drive away. 38For I have come down from heaven not to do my will but to do the will of him who sent me. 39And this is the will of him who sent me, that I shall lose none of all that he has given me, but raise them up at the last day. 40For my Father's will is that everyone who looks to the Son and believes in him shall have eternal life, and I will raise him up at the last day."
41At this the Jews began to grumble about him because he said, "I am the bread that came down from heaven." 42They said, "Is this not Jesus, the son of Joseph, whose father and mother we know? How can he now say, 'I came down from heaven'?"
43"Stop grumbling among yourselves," Jesus answered. 44"No one can come to me unless the Father who sent me draws him, and I will raise him up at the last day. 45It is written in the Prophets: 'They will all be taught by God.'[d] Everyone who listens to the Father and learns from him comes to me.
Is the source of grief for many due to its misunderstanding. Unfortunately instead of dealing with the context of His words, those hunting for proof texts to fit a theological grid have lifted them from their surroundings and placed them in the midst of their theological edifice (so they believe it to be anyway).
Their method is to first remove and then impose upon the words that which is not present. Take the statement "No one can come to me unless the Father who sent me draws him".
What does this mean? Just what it says. It is a testimony by God a declaration by God that man is in need of God's help and direction to find him. God must draw that man, all men. However, beyond what is clear is what others wish to force here and that is the concept that the reality that men must have God's providential draw, His Divine help in coming to Him, is that some how this means God chooses who will and won't be saved. Nothing in the text suggests this at all. Nothing. Zero. It is a conclusion drawn from supposition and not the text. It is an imposition on what is present.
But this is how the Calvinist is taught and exampled; to approach the Bible with such the expectation and demand to find that which supports their system of theology though texts fail to reasonably support it. And in that search any even seeming inference is seized upon and immediately enlisted as an aid to their theology though in reality a verse or passage simply stands indifferent or contrary to Calvinism.
Verse 37 is even more perplexing for some.
"All that the Father gives me will come to me, and whoever comes to me I will never drive away."
Instantly the Calvinist has been programmed to see and accept this automatically means "ELECTION". The first and most obvious question is by-passed, rendering the textual injury to the Calvinist.
To our Lord, Jesus, whom does the Father give? Those that believe on Christ as Savior. Believers are the Father's gift to the Son. Hence, all that the Father gives Him, will certainly come to Him and He (Jesus) will never drive away. Does God give unbelievers to Christ? Of course not, but this obvious question must be asked.
The point here is not election, as some hope in their claim but the certainty salvation in Christ.
And here is where the context is critical. Previous to the passage I quoted is the event. The event of feeding the 5,000 and the subsequent hunt for Jesus. Those looking for Him find Him and our Lord makes it clear:
26Jesus answered, "I tell you the truth, you are looking for me, not because you saw miraculous signs but because you ate the loaves and had your fill. 27Do not work for food that spoils, but for food that endures to eternal life, which the Son of Man will give you. On him God the Father has placed his seal of approval."
28Then they asked him, "What must we do to do the works God requires?"
29Jesus answered, "The work of God is this: to believe in the one he has sent."
The context is them wanting a kind of eternal life without truly desiring the truth. The response by them shows this:
30So they asked him, "What miraculous sign then will you give that we may see it and believe you? What will you do? 31Our forefathers ate the manna in the desert; as it is written: 'He gave them bread from heaven to eat.'[c]"
32Jesus said to them, "I tell you the truth, it is not Moses who has given you the bread from heaven, but it is my Father who gives you the true bread from heaven. 33For the bread of God is he who comes down from heaven and gives life to the world."
34"Sir," they said, "from now on give us this bread."
35Then Jesus declared, "I am the bread of life. He who comes to me will never go hungry, and he who believes in me will never be thirsty. 36But as I told you, you have seen me and still you do not believe. 37All that the Father gives me will come to me, and whoever comes to me I will never drive away. 38For I have come down from heaven not to do my will but to do the will of him who sent me.
This is not about establishing some mysterious doctrine of God only electing certain people to be saved. This is about the spiritual blindness of man in His sinful and fallen state and the nature of redemption. And here it is particularly emphasized because our Lord was dealing with the children of Israel whose claim to salvation was still tied to their national identity and through doctrinal ignorance had gone from understanding the prophecy of the coming Messiah 500 years before to now believing their Jewishness and their religious forms of righteousness could save them.
(Bible quotes from the NIV)