Why do many people say that Koine Greek is close to Modern Greek and distant from Attic, while grammatically it seems to be very close to Attic and still some significant distance away from Modern Greek?

Well has Dimitra Triantafyllidou’s answer put it:

Is the glass half-full or half-empty?

Here’s some ways in which Koine is closer to Modern Greek:

• Phonetics: there’s lots of disagreement about precise dates, but in lower-class Koine, potentially as few as two sounds were left to change over between Koine and Modern Greek, ɛ > i (η) and y > i (υ, οι). Accent was already likely stress- and not pitch-based, and vowel length was lost.
• Morphology: No dual, moribund optative. No Attic declension.
• Syntax: At the very start of hína replacing infinitive
• Lexicon: Substantial move forwards in both meanings of words, and Latin loans. Some of it straightforwardly legible by Modern Greek speakers.

Here’s some ways in which Koine is closer to Classical Greek:

• Phonology: Gemination was still present.
• Morphology: Still has dative, perfect, future, infinitive, third declension, athematic conjugation
• Syntax: Still has clause-chaining strategies using participles
• Lexicon: Still basically legible for someone reading ancient Greek

Phonetically, it’s almost Modern Greek. Morphologically, it’s identifiably Ancient, though there has already been some simplification. Syntax and lexicon are in between.