To what extent may head scholars of Ancient Greek language communicate with Modern Greek natives?

They can't communicate really.
Ancient Greek is very different from Modern Greek in terms of vocabulary, grammar and syntax. Reading an ancient greek text an average native Greek may recognise certain words or even make out certain phrases if they're well educated, but given the fact that the pronunciation has also changed I seriously doubt there could be any meaningful communication. Maybe if you stumble upon a language teacher or student.

What Erik Van Thienen mentions about the difference of Ancient from Modern Greek compared to Middle from Modern English I believe is a widespread myth (I have heard of it too). For example I, as a non-native speaker of English can read Shakespeare's works and understand them to a great extent (I have already read several plays). On the other hand as a native Greek speaker with a University degree (though not in Language) I cannot understand an Ancient Greek text of the 5th century BC apart from a few words although I was taught Ancient Greek for 2 years during high school. And if we're talking about spoken words in an Erasmian pronunciation, they would sound completely unintelligible to me even if they're words we still use today.

If you're interested in traveling to Greece though you should know that nowadays most Greeks speak english, especially the younger ones and even more so in tourist places, so you wouldn't have any trouble communicating.

Reasonably well, actually. There are more differences between Middle English (12th century - 15th century) and Modern English, than between Ancient Greek and Modern Greek.