Hi there @jjbinkes! Great to see your here at the forum, I hope you'll enjoy your stay here =)
I must confess I haven't read 'The Song of Achilles' and it's been a while since I read (parts of) the Iliad, but you make interesting points. It fits quite well, doesn't it? If memory serves, Achilles was a lot kinder towards Patroclus than he was towards others, which sounds suspiciously familiar... Achilles' heel indeed.
Looking at the Iliad, Achilles' wrath over being dishonored by Agamemnon is a central theme, which kind of ties in with Nezumi's will to take revenge on the city. (In which case that would make Agamemnon No. 6?) I think the gods meddle in at Achilles' request as well, hm...
I'm actually not sure, but my memory seems to insist Patroclus, until he went on his rampage, wasn't really considered a fighter at all, or at least a good fighter (or well, not as good as Achilles, but then again, that's Achilles). Which you could relate to no-one thinking Shion would harm anyone - until you hurt Nezumi, that is.
Well, it doesn't match up 100% of course, but those are certainly interesting parallels~ I wonder if reading the manga will lead to you coming up with more, since it's slightly different than the anime in places (I think I can state that much without spoiling ^^).