Actually, when koine Greek was first being critically examined, it was theorized that it was hardly Greek at all, but more of a hellenization of Aramaic. That was, of course, found not to be the case.

Yes, it is true that there is much structure in common between Koine and Classical, use of conjunctions, general grammar constructs. But for any real use of the language you quickly get into vocabulary domains that just do not intersect, or where they do intersect, the usage is quite different.