You have set a very high bar for yourself, as many language pedants do. I won't be stalking you, but I will be giggling in quiet as you make mistake after mistake, where the mistakes have been defined by none other than you.
I have not demanded perfect grammar from anyone, and when typing long sentences on a phone it's easy to make grammar errors that are too much of a hassle to correct. You are engaging in a pointless ad hominem.
The capital of my adoptive country is know locally as Athina, as it would be rendered in English. Is Greek culture or language somehow damaged by English speakers calling it Athens?
The vowel shift is a shift in Greek, not in English. Eta was a long epsilon
(roughly) prior to middle Greek. The differing ending makes sense since English is not a strongly inflected langugae like Greek.
The ancient theatre near Corinth (Korinthos) is called Epidaurus in English, rhyming roughly with dinosaur. In Greek it's Epidavros, which is very different.
Yes, because English generally takes Greek names from Graeco-Latin, which has different emphasis rules (typically, emphasis on third-to-last vowel). This is pretty consistent in how Greek names are rendered in English (AriSTOtle vs. AristoTEles, and so on).
While English does not render Latin (I think for practical purposes, we can group ultimately Greek words under Latin borrowings in English) perfectly consistently, there are broad patterns, and inconsistencies are generally attributable to vowel shifts. While I think it's to the detriment of English that spellings have not been revised nearly enough, these questions are much broader than a single particularly terrible romanization stumbling into the language, when similar words (other Turco-Mongol warlords' names) are rendered according to standardized rules.
If you don't see, however, how vehemently defending (while claiming not to be defending) a particularly terrible outlying representation of a foreign word with "It's the English name!" (which is circular reasoning in a normative discussion) is a form of cultural erasure, then you probably don't share the analytical framework I (and many scholars who study these things) use. To you, rendering a Mongolian name of a Mongol (who is the key figure in Mongolia's cultural heritage) in a way that completely ignores the Mongolian language is evidently value-neutral. To me, it's not, and being reasonably faithful to and respectful of the original languages has value in and of itself, and actively choosing
not to do so is a value statement.