Throughout the novel, Nick is the observer and impartial judge. He is related to the Buchanans and is a neighbor to Gatsby. He goes with Tom to see Myrtle, and he arranges a meeting between Gatsby and Daisy. He sees Gatsby as socially awkward and is suspicious of his past. He sees Tom as having a "cruel body" and living in the past glory of his college school football days.
But as the novel closes, Nick begins to take Gatsby's side. He recognizes the Buchanans as careless people, and that Gatsby is
worth the whole damn bunch of them put together.
After the "holocaust," Nick further distances himself by picking up the "mess" made by the others. Even though the responsibility of Gatsby's funeral is somewhat thrust upon him, Nick does a very conscientious job of contacting friends and relatives who would be interested in attending Gatsby's funeral. He even writes Wolfhshiem.
As Nick continues in the arrangements he begins to have
a feeling of defiance of scornful solidarity between Gatsby and me against them all.
Earlier Nick claims that
I found myself of Gatsby's side and alone.
Arranging Gatsby's funeral sets Nick apart from the Buchanans and the guests at Gatsby's parties. By being responsible, Nick shows his character and prepares the reader for his return to the Midwest where he would like to command the world
to stand at a sort of moral attention forever.
Chapter 9 actually begins 2 years later with Nick looking back at the rest of the day and commenting on what happened. You are right to indicate that Nick took responsibility for everything and the funeral arrangements. What is interesting though is that this was not necessarily by choice. Note what Nick says about how he was left with the responsibility himself:
I found myself on Gatsby's side, and alone. From the moment I telephoned news of the catastrophe to West Egg village, every surmise about him, and every practical question, was referred to me. At first I was surprised and confused; then, as he lay in his house and didn't move or breathe or speak, hour upon hour, it grew upon me that I was responsible, because no one else was interested - interested, I mean, with that intense personal interest to which everyone has some vague right at the end.
What is telling about this quote is that it was only in his life that Gatsby had "friends" - it was his wealth that attracted his companions to him. Obviously now that he has died and there is nothing to be gained from associating with him, nobody is interested, and Nick is left to organise the funeral by himself - a funeral to which hardly anyone comes. It appears that Gatsby, isolated in life, is never more isolated in his death. The fact that Nick organises his funeral serves to strengthen the link between them and emphasises the fascination Nick had with his character.