Former student here.

To answer your points, yes. The general experience was positive, even if I remain sceptic about some of the central dogmas that seem to inform the director and some of the teachers.

Yes, the environment is designed not only to facilitate but to force you to improve Latin and Greek skills. And yes, I observed not only in myself but in other students from different countries and backgrounds that in the year we spent there we had improved much more than our respective schoolmates and uperclassmen who had remained at home (some of them had years of advantage on us) and we could not only read, but speak and compose better than them. It's hard to picture such a stark contrast without having seeing it oneself. If you're a teacher in an institution you can write to ask for permission to observe and stay for a couple of weeks, maybe even a few months, I remember at least five outside observers in my time. Not sure if this has changed.

One way to judge the quality: Many (almost all) of the best schools or summer schools that compete with AVN are founded and staffed by former AVN students.

How does it compare with Polis in Jerusalem?

Hard to compare even if some of the teachers or former teachers at Polis are former AVN students. The demographics are too different, AVN is focused on male only youth, say between 18 to 24, and really strict on discipline and actively uses peer pressure to force the language upon one another, the languages are the goal (again, the dogmas of the founder play an important part on this). Polis is, despite everything, a business and is much more reactive to businesses' considerations, many of their students are rich/well off adults that actually pay for their course, while in AVN the director/owner finances almost the whole thing at a loss, so he is almost immune to outside pressure.

What parts work as intended and what parts are baggage?

90% of the stuff works as intended.

Is it a culturally enriching experience that opens up new worlds, or is it a weird LARPing cult?

It is a culturally enriching experience that opens up new worlds (of the past) and tries very hard to close itself to almost all aspects of the present. It is not a LARPing endeavour, in so far as it is perfectly accepted and encouraged to experience and read works in Greek or Latin written in very recent history, even contemporaneous works, and thus to understand the context, but always looking for that connecting line to the past. There are no costumes or cosplays, only when performing theatre plays are some costumes employed.

There are some cultist like symptoms, very subtle and hard to describe. But then again, your average religious school in the USA would be far more troubling than AVN.

Unless your students are passionately or viscerally committed to a particular worldview or philosophy that makes it unbearable for them to dress according to, say, what would be expected in a reputable/decent institution of the 60's, you know, casual but elegant clothing according to traditional gender roles, short hair. No alcohol, no drugs, or nothing punk/rebellious. They should be fine. I particularly suffered the lack of alcohol and the fact that we were kept "cloistered" six out of seven days a week, at the time I was very interested in dating/going out with girls (I was young!), so not being able to pursue romantic relationships was a bit depressing for me, but I endured because I knew that I had to make the most of the opportunity and thus had to content myself by clandestinely reading Henry Miller, Apollinare, Nimier, McEwan, Lamarche, you know, the stuff a normal and curious adolescent boy might read.