Chapter 2: Topical Index
In this narrative, I believe it is very clear that the primary purpose for this whole scenario and this particular account is given to us in vs. 2:6-11. The purpose of this miracle, Jesus' first, was to manifest His glory and bring his disciples to saving faith.
This beginning of miracles did Jesus in Cana of Galilee, and manifested forth his glory; and his disciples believed on him. (John 2:11)
I agree when you say that the purpose of the passage is plainly intended to show us how his disciples come to a belief in Jesus. Coming to see Christ as God through his significant miracles--or, miracles which signify--is a significant part of the Gospel of John, showing us again and again the proper way to 'see' Jesus, beyond what our earthly eyes would otherwise reveal. It is also enlightening to examine the various ways in which John prompts us to see Christ through his gospel narrative; one of the reasons it is so different from the synoptics in structure and style is, I believe, the very distinct and clear agenda of the author, which is stated bluntly at the end of the gospel.
And many other signs truly did Jesus in the presence of his disciples, which are not written in this book: But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name. (John 20:30-31)
Examining John with an eye towards the very specific selection of miraculous events can open new depths of meaning to the gospel; it is truly not an exact science, not the kind of thing we can measure by the standard of factual information by which the world works; that is, to my limited way of thinking, a different kind of sight, a different kind of evidence, and while the outward appearance of a verse is very revealing, a peak into scripture's inward meanings can shed light on the author's intention, and has frequently challenged me to a more fruitful discipleship.
Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me. If ye had known me, ye should have known my Father also: and from henceforth ye know him, and have seen him. Philip saith unto him, Lord, shew us the Father, and it sufficeth us. Jesus saith unto him, Have I been so long time with you, and yet hast thou not known me, Philip? he that hath seen me hath seen the Father; and how sayest thou then, Shew us the Father? Believest thou not that I am in the Father, and the Father in me? the words that I speak unto you I speak not of myself: but the Father that dwelleth in me, he doeth the works. Believe me that I am in the Father, and the Father in me: or else believe me for the very works' sake. Verily, verily, I say unto you, He that believeth on me, the works that I do shall he do also; and greater works than these shall he do; because I go unto my Father. (John 14:6-12)