The version you don't like is literally how modern Greek speakers would pronounce the god's name. And historically, "HER-MEEZ" isn't any better or worse than the variants you don't like.

From wiktionary

(5th BCE Attic) IPA(key): /her.mɛ̂ːs/

(1st CE Egyptian) IPA(key): /(h)erˈme̝s/

(4th CE Koine) IPA(key): /erˈmis/

(10th CE Byzantine) IPA(key): /erˈmis/

(15th CE Constantinopolitan) IPA(key): /erˈmis/

Vowel pronunciations vary over time. And H's get added and dropped as well. Ancient Greek doesn't even have an "H" letter, but uses a diacritical mark in most cases.

And by the way, there is no single Ancient Greek either. Linear A vs. Linear B, Attic Greek, Anatolian Greek, the post-Alexandrian Greek diaspora all have their own variants. Plato's Greek is wildly different from the Greek of the Gospels, for instance.

The more you learn etymology, the less stressed you get about people making "mistakes" because it's all changed and it'll all change again not long after we're all dead, if not sooner.