So the most important thing here is that the story of the Trojan War is just that -- a story. The idea that the story Trojan War accurately reflects conditions of the Bronze Age is one that hasn't been widely accepted in academic circles for decades, despite the fact that it's still commonly assumed in popular media. Examples of the latter include the movie Troy (2004) and the game A Total War Saga: Troy, released last year.

Some part of the story of the Trojan War may well date back to the Late Bronze Age, but definitive proof of knowledge of the story of the Trojan War doesn't show up until the Archaic period of Greek history. The earliest texts are the Iliad and Odyssey, which are attributed to Homer (whoever that might have been), and are dated to ca. 700 BC or possibly a little later. It is, indeed, not entirely clear when exactly the texts were written down: an ancient tradition suggests the texts weren't written down more or less definitively until the sixth century BC.

And as far as ancient Greek art is concerned, there are no depictions of anything that can be positively identified as representing a scene from the Trojan War until the seventh century BC. The definitive study in this regard is Anthony Snodgrass's Homer and the Artists: Text and Picture in Early Greek Art (1998). Snodgrass shows that when Greek artists did start to depict scenes from the Trojan War, they often didn't chose things we'd recognize from Homer. It's good to remember that many stories circulated orally and that the Homeric epics were part of a bigger collection of poems referred to as the Epic Cycle.

So you're question is actually up to three different ones, relating to the problems associated with mustering and leading a large coalition force in:

1. The Homeric epics.
2. The Mycenaean Bronze Age.
3. Homer's own time (presumably ca. 700-650 BC).

These are questions that I have answered before, so I hope you don't mind if I direct you to threads that I think are relevant here. They are:

• Are most Ancient Greek myths “set” in Mycenaean Greece? (This answers the question of whether or not ancient Greek myths -- including the Trojan War -- relate in any meaningful way to what we now refer to as the Bronze Age. The book by Boardman that I cite in my response is very useful for understanding how the ancient Greeks actively created their own past.)
• What was the relation between the mycenaean(17th-12nd century BC) and the ancient(8th-2nd century BC) greeks? (For context, I think this will be useful.)
• In the Bronze Age Mediterranean, states such as Hattusa, Egypt, Assyria, et alia are described as unified, unitary monarchies. However, I have not seen any in-depth exploration of the government of the Mycenaeans. What sort of state(s) did they possess? What forms of government did they practice? (This contains a lot of relevant information on how the Mycenaean palaces were organized, and also concerning the historicity of the Trojan War.)
• At the time the Iliad was composed, was it really common or possible for a city to be held under siege for 10 years? Is the nature of the war as described familiar or fantastical to its audience? (I talk about the scale of the war; relevant to the question you asked.)
• Military structure and warfare of Ancient Egypt? (We know a lot about the ancient Egyptian military, so you might want to refer to this by way of comparison.)

Finally, you might also want to check out the AMA /u/Iphikrates and I did about the Total War game. It goes into some depth about the topic. Among other things, I answer a question about how raising armies worked both in the Homeric epics and during the Mycenaean Bronze Age (you can CTRL + F for "Raising an army: the Mycenaean palaces" to find it more quickly).

Of course, feel free to post follow-up questions you might have.