Take this with a bucket of salt because I haven't read much on this topic.
But just reasoning 'rationally', I assume the argument is that the Iliad / Oddysey were told in cultures of predominantly oral tradition? So likely, just as with the game of telephone, the story got told and retold, and distorted, many times until someone ("Homer") decided to write it down?
So the argument being that Homer is not the 'creator' of the stories, and might just be someone who wrote it down?
Or perhaps the argument is that no single person wrote it down?
There are some plausible arguments that the authors of the Iliad and Odyssey are not the same, even if they must have been closely related.
For instance I find very plausible the hypothesis that the Iliad was composed by a man and the Odyssey was composed by a woman.
The main reason is the striking resemblance of the Iliad with an action movie, in contrast with the Odyssey that looks exactly like a chick flick.
More seriously, the Iliad focuses on a lot of things of great interest for the men of those times, like gory details about the best methods to kill or maim opponents in combat, or athletic competitions, while the Odyssey focuses on many things of interest for females, like clothes, food, gardens, romance, and it even has several passages that look quite feminist, despite being written millennia ago, by complaining about the discrimination unjustly enforced by men against women. The Odyssey also has a lot of strong female characters that are more important than the men, e.g. Calypso, Circe, Aurora, Nausicaa, Penelope.
The text of the Odyssey also contains evidence of being composed later than the Iliad, perhaps by several decades (due to some evolution of the language). So it has been hypothesized that the Odyssey was composed by a female relative of Homer, e.g. by his daughter or niece.
In any case, the author of the Odyssey mastered perfectly the same kind of language and poetic structures that were used in the Iliad, so he or she must have studied for many years the older poems in the same tradition, like the Iliad and the many others that have been lost.
A very large number of verses and short sequences of verses from the Iliad and the Odyssey are much older than the complete poems. However, someone, i.e. "Homer" alone, or with his assistants or relatives, had integrated all the older inherited poetic fragments into big coherent epic poems, before they were recorded in writing.
Aurora is the Latin name of the Goddess of the Dawn, one of the very few gods that have been inherited since Proto-Indo-European times. After migrations, the Indo-European people have adopted most of their gods from other populations, preserving only a few of their ancient gods, like the God of the Sky, Greek Zeus = Latin Jupiter and the Goddess of the Dawn, Greek Eos = Latin Aurora. Despite superficial differences, the Greek names and Latin names of these god and goddess are the same, descending from a common form through phonetic alterations (Dieu => Zeu- = Ju- and Ausos => Eos = Auror-, through regular phonetic changes, while -piter means father and -a was added to Auror to make it more similar with most other female names).
In the Odyssey, there is a lamentation about how the female goddess Eos = Aurora is strongly criticized for taking human lovers whenever she wants, while when the same thing is done by male gods, like Zeus = Jupiter, that is overlooked or even praised. Therefore this criticism about how promiscuity is treated differently by society for females and for males is millennia old.
You have a point that the Odyssey also mentions athletic competitions, not only the Iliad, but the importance and the amount of space dedicated to describing such events is much less than in the Iliad, where an entire chapter is dedicated for this and there are very detailed descriptions of the techniques used by each competitor.
In the Odyssey, the athletic competitions are just a vehicle used to highlight the physical abilities of Odysseus, demonstrating that even if he is older he can still easily outmatch younger competitors.
In the Iliad, the athletic competitions are described much like a radio or TV commentator would describe the events to spectators, focusing more on the description of the actions of the competitors as a show of skill, than on the results.