You are correct about the lack of consistency in Greek myth, which, after all, was set down over a considerable span of time. There was no need for it all to be consistent, and it would’ve been hard to coordinate it all, across that large an area (Greek civilization extended across not only present-day Greece, but multiple islands, a big chunk of modern Turkey, colonies in Sicily and Italy, and elsewhere). Furthermore, there was no compelling reason to – different regions had their own heroes, and their myths tended to emphasize their own guys over those of elsewhere. That the Greek myths ended up being as cohesive as they did is pretty incredible, given that.
Nevertheless, there are a LOT of inconsistencies and contradictions in Greek myth – my list above barely scratches the surface. Pull out a book that goes in-depth on Greek myth – Karl Kerenyi or Donald Ganz or Robert Graves*, or a majestic tome like Roscher’s Lexikon – and you’ll see all the variations and inconsistencies. Those genealogies they often put in the backs of books on myth need to be taken with a grain of salt.
It was much later, circa the first century, when (pseudo) -Apollodorus started collecting the myths into a single volume, and wanted to try to fit them together into a coherent non-contradictory whole, that the began hammering out the annoying inconsistencies that stuck out, and rationalizing them away. Theseus encounters a Gorgon in the Underworld? Aha! It must be Medusa, killed by Perseus, because the other two are immortal! And Theseus’ visit must therefore be after Perseus killed her.
What’s really amazing, to me, is that you have all these stories about the way the Universe is set up, and important events in World History, and you leave it in the hands of Bards and Poets to recite, compose poems about, and propagate. Could you see the Catholic Church letting Andrew Greely decide what Catholic doctrine was by writing it up in one of his novels? Yet that’s exactly what Greek religion effectively allowed Homer, Hesiod, Pindar, and that crowd to do. Pretty amazing, when you contemplate it – because a poet will be driven by what fits metrically, works intuitively, and ends in a dramatically satisfying fashion. He or she isn’t compelled to follow what is believed to have actually happened. (As I’ve suggested above, in the case of Achilles/Agamemnon’s shield, Art tradition can diverge from literary tradition. The myth of Pandora looks completely different on Greek pots than in Ovid’s familiar version. There are illustrations of Jason that don’t at all fit in with Apollonius of Tyre’s poem, and so on.)
*whatever you do, though, take Graves with a skeptical eye. The man notoriously tried to pass off his imaginings and suspicions as reality, without clearly marking them.