[A_Brandybuck]

Beside the point, that the Fellowship is a picture of the whole humanity, I have another theory.

There is not the classification, where one character represents one race. This isn't possible, because of two Men and four Hobbits. When we see from another point of view, Hobbits could be classifed as Men, because they are a branch of Men. Consequently six Men, one Dwarf, one Elf and one Istari as the leader.
This fellowship represents in a way the actual status of the free peoples of Middle-earth. The time of Men has come, the Elves have not more than one in the Fellowship and this one isn't one of the Noldor. The Dwarves have also only one ambassador, but the Dwarves are more interesting in their own concerns than in the concerns of Middle-earth.
The leader is an Istari, who came over sea to help and motivate the free peoples of Middle-earth, which are probably represented by the Fellowship, which he lead. The Istari haven't come to make war against Sauron, they wanted to increase the will to fight against Sauron in the hearts of the peoples. Do it yourself would be a sloppy formulation.

Exactly this happens after Gandalf was gone (or fallen). A Man took over the leadership, not an Dwarf or an Elf. (An Elf could be the first choice in former times) A

That is what I think about the representation of the Fellowship. I hope that isn't too much skurrile or too far away.