I think the problem stems from treating languages like koine and latin as "dead" languages.
At the same time, it would be anachronistic (or just pig-headed) to say that modern Demotic Greek is the same thing as Homer's verse, Herodotus' history or Plato's prose. Spelling, morphology and syntax in each of those dialects are different (even compared to katharevousa
), and the first category (spelling) shows us incontrovertibly that so too was pronunciation. (What that pronunciation was exactly
is another matter!)
At any rate, we now have epistolary evidence from Hellenistic Egypt that documents several shifts in pronunciation (e.g. mildly literate people will consistently misspell words by using an eta where there should be a iota -- or some such thing -- a reading that indicates that the conflation of vowel sounds dates to a fairly early period, at least in the diaspora!).
For what it's worth, some of the best scholars I know over at Harvard insist on Byzantine (i.e. modern) pronunciation in their courses, although they work heavily in the manuscripts, so they have an appreciation for Byzantine Greek.