I've read Hamlet's Mill, having been given it out of the blue by a nice guy who worked in a second hand bookshop in Cirencester. I found it very interesting, but it didn't trigger my doubts.
The eclipse on Ithaca paper was much more recent, and seeking to find an actual date for Odysseus' return, and I was prepared to consider it until they started to use Mercury. The idea that you could prove the existence of Odysseus as a real person who did things on a real date seems a bit wild.
Other things I have read which tried to use astronomy for odd purposes were a pair called "Homer's Secret Iliad" and "Homer's Secret Odyssey" by F & K Wood.
The Iliad, apparently, is entirely set up to transmit the knowledge that Orion becomes invisible for a period of time, Orion being represented by Achilles. I went along with this, given the annual cycle of the heavens, until they revealed that what they were on about was that for a period during the precession of the axis, Orion would not be visible in Greece at all. That Homer would know this, and then decide to bury it in something which to all intents and purposes is about human behaviour is stretching things beyond what they will support.
The Odyssey is worse. This, apparently, is describing the movement of the Sun throughout a year. Since, at three occasions during the hero's journey, he engages in retrograde motion, which, unless you are living on Mercury, the Sun never does, this is daft. I actually went to the lengths of printing out star maps and tracking what the Woods said the Sun did in visiting various constellations - not only retrograde, but also way off the ecliptic, and full of zigzags north and south.
I am left with the question "Why"? Why would anyone in the past have gone to the described lengths to pass on the "secret" knowledge? Why would it have been seen to be important? And why would anyone now want to spend the time investigating this. And why would any publisher publish it?
I actually think that people composing a work of fiction have an idea of where they want it to go and then add into it picked up unconsidered trifles (reference to character with the name of Odysseus' thieving ancestor) which add interest to the narrative, and perhaps to the character they are dealing with. They don't have to be actual experiences from an actual time. Ah, an eclipse, that will add drama to the return. A volcanic eruption, that will make a different sort of storm to destroy the ship.