Should be noted- i think it is revealing - that anyone speaking current greek can easily read the new testament (which- as known- was written in greek originally). That doesn't mean the language in the new testament was particularly refined (it wasn't). Yet it does help show that the greek language is continuous since ancient times, or at least if you go up to the first century AD or even up to Aristotle (who is closer to current greek than Plato is, while Homer is impossible to read if you do not know actual ancient/archaic greek he used).
Yet even in Homer, most of the words used still exist, albeit most of the time they have acquired different meant. Homer's grammar is not close to current greek grammar, though...
As a final note, i had some difficulty reading Byzantine-era manuscripts. It can be done, although again the same words have different meaning. The grammar is virtually the same, though. And the 'kathareuousa' you mentioned, yes, it was a more rigid version of greek, yet naturally it got a sort of fusion with more usual grammar (less austere) used up to now. Then again, the counter-movement (in the 60s iirc) to have a different and simplified version, the so-called 'demotike', followed the opposite direction, given what is now used is a mixture of the two.
I think at least some of that is to do with the pressure, for about the first century and a half of Greek independence, on people to see increasingly 'ancient' forms of speech and writing as increasingly 'correct'. In other words, as you've said, just as
katharevousa took on features of
demotike over time, so the regional and modern features of
demotike declined throughout the 19th and 20th centuries, as well as some deliberate official attempts to bring features of
koine (New Testament) Greek into the everyday language. Interestingly, the law phasing out
katharevousa as the official language said that the new one would be
demotike, but 'without regional or extreme forms' - which ruled out one popular idea of
demotike as a language which should embrace all of these, as the true language (in the sense of what they actually spoke) of the Greek people. One particularly important contribution of
katharevousa is in how modern Greek handles words that weren't around in Antiquity. It normally forms its own compounds as an ancient speaker would have done, rather than (as, say, French and German usually do) borrowing the word directly from another language. Hence 'television' is τηλεόραση (far vision), a car is an αυτοκίνητο ('self-mover') and computer is
ηλεκτρονικός υπολογιστής (electronic calculator).
It's less straightforwardly 'continuity', and at least part a process of conscious reconstruction. It would be interesting to have gone around the villages in the final years of Turkish rule and found out how many of them could read Plato.
This (sort of) moves towards a second point - that the similarities are strongest when the two languages are written down, because the Greek writing system is (and historically was) extremely conservative. Modern Greek uses the same characters as Ancient Greek to make totally different sounds, and has a completely different system of accenting words - though until the 1970s, was still written with the ancient polytonic markings, which hadn't reflected ordinary speech since (at least) the Byzantine period. If you wrote ancient and modern Greek in the same system (like the IPA, for instance), you'd notice quite a lot more differences than are obvious on the page. For a modern speaker, understanding the manuscripts of Homer might be one thing, but it would be another thing entirely to have understood somebody reciting them.