Hans
It is possible to recruit authentic material into literary device. Thus, poetic treatment of a historical theme doesn't necessarily imply totally imaginary history. Are there literary devices in Homer? Was there ever a Trojan War?
1. But was there ever an Achilles? Well, probably nope. (Even the name is not a Greek name, but basically a literary role description.) Was there ever a king Agamemnon? How would Homer even know of him? Was there an actual Odysseus?
2. Occam's Razor. Between an explanation with one entity, and an explanation with two entities, take the former, unless you actually have evidence that requires the latter. If its being used in an allegory explains the presence of an incident well enough, then you don't need an explanation which requires BOTH that AND a historical incident that incidentally is an allegory and foreshadow. Unless you actually have evidence that requires the latter, that is.
3. Most people's lives don't follow such neat narratives. You'd be hard pressed to find many real people whose lives even follow a rough monomyth at all, much less involve half a dozen foreshadows for every major event, and where whole events are neatly arranged to form a layer cake of nested allegories, half a dozen levels deep. When a narrative of someone's life ends up being composed of such, that's usually a hint that some creative rewriting has happened. Which leads us to:
4. How would you know which parts, in the absence of other evidence? How would you know which allegories are freak cases of actual events that are also allegorical, and which are just made up?
5. Most importantly because it makes the following false:
Surely, too, the principle of inconvenience still applies.
No. If the whole incident serves to make a point that the author wants made, then there is no inconvenience.
Nevertheless, if there is a historical basis, however slight, surely it is to be found in these inconvenient locations.
Not really, or not necessarily.
The criterion of embarrassment is actually used to guess which of two versions of the same story is earlier, and which is later, not to determine which parts are real. E.g., if you find two versions of a gospel, say, one where Jesus shows compassion to some sick person who touches his robe to get healed, and one where Jesus gets angry at the sick person for touching him, the heuristic guess is that the less pleasant version is earlier.
The idea is that as the manuscripts would get copied again and again and again and again [...] some scribe would notice that some details aren't all that popular or convenient, and change them to something more palatable. In the process of copying, the rougher edges and kinks were also eventually ironed out. So when you find a version with more kinks, chances are it's earlier.
But it's not really a criterion for finding historical truth. An earlier version can have those quirks simply because the author wanted it that way, or his particular audience liked it that way.
Think the various versions of Little Red Riding Hood, or Hansel And Grettel, or various other children's stories. Earlier versions were a lot nastier. E.g., LRRH is killed and stayed dead even in the relatively late Perault version, and even earlier versions have even more gruesome stuff like the girl being fed pieces of her dead grandma. Yeah, nasty stuff. E.g., even in the first Brothers Grimm version, H&G were getting the crap end of the stick from their biological mother, not from an evil stepmother, and there are indications that in even earlier versions, the evil witch was also their mother. But as time goes by and the story has to appeal to larger audiences, the versions of the story become increasingly toned down, and the nasty parts get increasingly ironed out. So if you find a nastier version of them, an educated guess is that it's earlier than a nicer version with more mass-appeal.
But it's not a real indication that there was a real LRRH, or that certain events in her life are historical.
Or take Odysseus, since you brought that epic in as an example. Odysseus's behaviour, like pretending to be crazy to avoid honouring an alliance, or coming up with cunning stuff like the Trojan horse, were extreme negative traits for the Romans. The Roman ideas of honour and valour were much more strict, and for example alliances had to be honoured at any cost. About a millennium of Roman theatre, poetry and novels have been dedicated just to vilifying Odysseus and his treacherous behaviour.
So, can we apply the embarrassment criterion to find historical stuff there? Not really, because for the Greeks it wasn't embarrassing at all. Winning wars by cunning or avoiding wars by cunning, were actually positive traits for the Greeks, unlike for the Romans later. There was no embarrassment.
What we can however say is that the story is earlier than the people in Rome who were annoyed by it. But then we knew that already for Homer.