Reddit reviews Ancient Greek scholarship: A Guide to Finding, Reading, and Understanding Scholia, Commentaries, Lexica, and Grammatical Treatises, from their Beginnings to the Byzantine Period

The short answer is yes, it's possible.

But the problem you'll run up against is the lack of adequate resources. First, you'll want Eleanor Dickey's Ancient Greek Scholarship, with its dictionary of grammatical terms and summary of the ancient grammarians. (Start with Dionysius Thrax if you want to try the original.) This is the work referenced by Annis on the document you found at scholiastae.org. You could also try to get a copy of Emiliano Caruso's Vocabolario monolingue di greco antico (but see the discussion at Textkit). Caruso is a decent start, but we have nothing as high quality as, e.g., LSJ in ancient Greek (nothing like Forcellini for Latin).

Probably the easiest way to try to keep Greek in Greek is to purchase materials such as Christophe Rico's Polis textbook, written entirely in Koine Greek. Rico's book will gradually introduce you to Greek grammatical terms. Also have a look at Randall Buth's Living Koine materials (though I don't know if he presents grammar explicitly).

Ultimately, the best thing to do is, if you can, to attend an immersion course taught by a competent speaker of ancient Greek. The Polis Institute offers such courses year-round (in Jerusalem) and during the summer (in Jerusalem, Rome, Boston, and elsewhere); the Paideia Institute offers one in August in Greece. Once you have a couple years of Greek down, you could attend something like the Σύνοδος ἑλληνική offered this year for the first time in Kentucky.

If none of those is possible, the Paideia Institute also has some online classes taught in ancient Greek.

In sum: it is entirely possible to study ancient Greek mostly in Greek. It will be slower in some ways than if you used mostly or exclusively English (vel sim.) as your medium, but it will offer the benefits you've already experienced. And the better you get at Greek, the easier it will become. You'll be able to start using resources like Gaza's Attic prose paraphrase of the Iliad and the ancient Greek scholia to many Greek texts, and so on.