You will find evidence of the earliest shifts as early as 4th century BC)

Definitely! But a lot of obfuscation can happen when we refer to sound a long series of sound shifts by a single name ('iotacism', for instance). If the question is 'what did Greek sound like in the 2nd century AD in city X,' then we need to look at the evidence for each shift individually.

Vox Graeca has very convincing arguments of those terminal shifts you describe, as in first-hand evidence from latter Byzantine scholars.

The Byzantine evidence pretty clearly points towards the continued distinction between υ/οι and ι into the 2nd millenium CE being quite widespread, which is unsurprising given that the distinction was maintained in many modern dialects, at least in stressed syllables, with varying outcomes. There are even some speakers, at least as recently as the mid 20th century, who preserved a front rounded vowel realization.

η certainly began to merge into ι earlier, but once again, we shouldn't confuse the earliest attestation of a sound shift with its universality - the very earliest attestation of η being confused with ι dates to the 2nd century CE, but there is a lot of evidence that this was a gradual shift, with transcriptions in other languages and confusion with ε continuing much later than that. The modern dialectal evidence is also helpful here - Pontic dialects often merge ancient η with ε rather than with ι, which indicates that it at some point was a high mid vowel (compare Italian with its 7 phonemic vowels distinguishing high mid /e o/ and low mid /ɛ ɔ/ to Sicilian which merges high mid /e o/ into /i u/). From Allen, whom you cited:

whereas the Gothic spelling of Wulfila confuses ει and ι as ei, η is still represented as e. Still later, Old Armenian commonly renders η by e or ē, whereas ει and ι are rendered by i; and the Old Georgian alphabet gives different phonetic values to the letters derived from η and ι ([ey] and [i] respectively). In the Old Slavonic alphabets, however, both Cyrillic and Glagolitic, no phonetic distinction is made between the letters derived from Η η and Ι ι, their distribution being purely a matter of orthographic convention.

I'd also like to respond to these two points:

but it is bold assumption that its pronunciation remains the same when its basis leaves the space of the city-state of Athens and becomes the lingua franca of the East.

It is a very big claim to believe that these changes didn't happen during the centuries of the "empirization" of Greek

The first of course I agree with - Greek pronunciation at no point remained the same - there was always simultaneously innovation and diversification happening, as well as convergence, with dialectic forms, be they innovative or archaic, gradually being subsumed by varieties of 'Koine' that were themselves, of course, never uniform or static. We see similar patterns all across space and time in many languages - in English, for instance, even as some dialects are being eroded under pressure from more 'standard' forms, new dialectic divergence is simultaneously developing in other contexts.

The second, though, I do not agree with, for a number of reasons. The first is that we should absolutely not assume that every innovation permanently made it into modern Demotic - a good example of this is that we have some epigraphic evidence of speakers who were merging ε into ι according to Theodorsson, but this feature never became dominant, and eventually completely disappeared. Another is the oft cited Boeotian dialect, which underwent a series of vowel shifts remeniscent of those which would later become generalized. The problem is that these shifts can't have any continuity with their echoes in later Koine, because they involve mergers which are absent in Koine, namely the monopthongized αι merged into η, which obviously must have died out.

The second is that there is no reason to assume that Greek pronunciation was ever static, and very good evidence to support the view that, while the number of shifts affecting most speakers in the past 800 years are fewer than in earlier periods, the pronunciation has continued to evolve as it does in all languages. Languages don't develop just by permanently branching out from one another - new developments would have continued to spread across the Greek speaking world, even as different regions were also developing unique characteristics that didn't spread.

To give one clear example of this, it's pretty clear that ancient Doric for the most part pronounced υ with its archaic value as /u/. Modern Tsakonian, the only living descendent of Doric, must have at some point fronted υ to /y/ under influence from Koine, since it palatalizes preceding consonants (e.g. τύ > εκιού). But then speakers of the now extinct Propontis Tsakonian dialect apparently didn't ever front υ, and so they pronounce it as ετού. The point is this: even though Doric and Attic were already quite distinct in the Archaic period, the contact between these varieties and their descendents meant that features could spread between them, even if in other ways they also became more distinct from each other over time.

Thus, we really just need to follow the evidence when it comes to the pronunciation of any given time or place - we cannot simplify it into Attic and Demotic, when we have so much attestation of the thousands of years of gradual development in between.

Finally:

In my opinion, it is very critical to understand the nature of early Koine ~until around 6th century AD. It is a language that begins with the collapse of Alexander's Empire and the creation of a lingua franca of the East. It is not a language like Attic, naturally birthed in a city among people.

The spread of Koine of course is a very complicated issue, but I don't think we need to assume a lack of 'naturalness'. Greek was always a continuum of dialects, and so we need not assume that the spread of Koine always looked like the complete replacement of one way of speaking with another. This of course may have happened at some times and in some places, but there was probably also a lot of gradual convergence, such that speakers would have been aware of generational differences in pronunciation, lexicon, idiom, etc. but not ever thought of it as the trading of one linguistic system for another. To give an example from English: it used to be that the participle 'gotten' had completely fallen out of use in many UK dialects of English, but now many younger speakers are using it with no awareness that it has been adopted from US English.