The connection to verse 20 doesn't
exclude verses 25-26 or 35-36. It explains what
triggered the entire discourse. The arrival of the Greeks in verse 20 is the narrative catalyst:
"Now among those who went up to worship at the feast were some Greeks. So these came to Philip... and asked him, 'Sir, we wish to see Jesus.'" (John 12:20-21)
John deliberately isolates this event: "Now there were some Greeks..." (δὲ marks a narrative transition). He draws attention to them as a separate narrative unit before Jesus speaks again. The significance of it is that this is the first explicit mention of Gentiles seeking Jesus in John's Gospel. Up to this point, Jesus' ministry has been almost entirely within Israel. There had been occasional foreshadowings of Gentile inclusion (e.g., John 4:42), but this is the first time Gentiles are physically present and requesting audience with Jesus.
Jesus "answers them" (ἀποκρίνεται αὐτοῖς) in verse 23, saying, "The hour has come for the Son of Man to be glorified." The Greeks' arrival is what provokes this declaration. It is the event that marks the shift from "My hour has not yet come" (2:4; 7:30; 8:20) to "The hour has come" (12:23). So the arrival of these Gentiles signifies the transition from a Jewish-restricted mission to a universal one. Everything that follows Jesus' declaration in v. 23 unfolds as His theological exposition of this turning point.
Verses 25-26 develop the implications of His glorification (the necessity of death leading to life), while verse 32 gives its climactic significance: He being "lifted up" will effect a drawing of all kinds of people: Jew and Gentile alike, not Jew only. That's why John mentions the Greeks at all. They are the narrative signal that the redemptive focus is expanding beyond Israel.
The logic of the discourse is:
vv. 20-23 - The Greeks arrive --> "the hour has come."
vv. 24-26 - The principle of life through death (the grain of wheat).
vv. 27-33 - The meaning of Christ's death: the cross as the means of universal (not Jewish-only) gospel appeal.
vv. 35-36 - The closing exhortation: believe in the Light while it is among you.
The meaning of "draws" (ἑλκύω) and the scope of those drawn are not determined by the exhortation. In this context, the verb concerns the inclusion of
all kinds of people; that is, kinds without distinction, not individuals without exception. Jesus is announcing the ingathering of both Jews and Gentiles into one redeemed people, not the universal salvation (or attempt at it) of every individual. Moreover, the semantic core of ἑλκύω is forceful or powerful, not merely inviting. The core idea it expresses is the
decisive movement from one state or sphere to another. Thus, when Jesus declares, "I
will draw all people to myself," He is not describing a mere attempt to persuade; He is proclaiming the certain efficacy of His redemptive work: the power of the cross to extend through the gospel to all nations and to bring people of every kind to genuine faith in Him. If that "drawing" is taken to refer to individuals without exception, the text would be teaching universalism.
Your analogy, hiring a pedophile to work in a daycare *_* , entirely your choice of words, illustrated my point, not yours as to which one was peculiarly unsuitable for office taking into consideration the crime and the office. 
