Hi mich
Really weird you ask this question today because I was having quite a moment about DS 5's summer camp. We had issues with his schools placement although not at first. He started at 2.5 and when signed him up at 2 I didn't know he was advanced, or as advanced. I didn't know what we did was unusual, kind of silly in retrospect, although his big explosion of development came between 3.5 and 4.5. So we chose "the best" place in our area. It was "academic" And in a lot of ways the school was great. In the 2.5 class it was 3 hours, 3 x a week and the day was really about playing with others, lining up, going to the bathroom, music, etc. The teachers were very attentive and caring.we got a lot of comments about "the things he says" they were much more
"adult" he would tell them about cooking shows and books and things, but also very elaborately. I don't think he was bored then but who knows he could really articulate the concept. He always wanted to go. Their bulletin boards were filled with the kids work and the introduced letters and numbers etc. But things change the next year
as the distance between his abilities and the other kids just grew and grew and as the
kids really moved from parallel play to playing together it became much more noticeable that what he liked to do and talk about was different than the kids. And this got much much worse in year 3, his pre k year. They were academic they did books, and letters and numbers and worksheets. But they also felt that DS social skills were
a problem and there goal was not to teach him letters or numbers, he was reading, but to get him to be more like the other kids. It was so disheartening and it took me a long time to realize that the teacher had no appreciation for the different box that DS was in. Fast forward today, new camp in the "best" nursery school in our new area and wow what a difference. They love his differences and rather than asking him to conform they are showing him how to handle when other kids don't understand him. I almost cried talking to the counselor.
I do think though there is a big difference between camp and school even in nurserry settings, DS went to camp at his old school too and it was run by different people and so had a different style. I hated the old camp, even if we hadn't moved we weren't going back because they didn't have trained educators in charge they had college kids. In his current camp, there are two head counselors who are teachers at the school and 3 college students who are in early childhood Ed as assistants. The difference is stark - the non early childhood Ed counselors didn't really know how to deal with the age, so possibly that is what you are seeing. Also probably why the boards didn't relate to what they were doing.
Your DS reminds me of mine in terms of the behavior, mine is very well behaved at school for the adults because he considers this the interesting time,its the gathering of new info which is what he lived for. So mine also sat and listened and would tell others to shush. And the following directions part, they look so different because of the following multiple step direction, most are still having trouble with that.
The buzz word I didn't know then but having been to the new camp and getting a feel for their school is individuality and unique - the new place seems to be about introducing age appropriate social concepts but by figuring out each kid they focus on what that kid needs to work on while providing fun. There is really development play here or play based learning rather than just play.
The boredom issue might always be there - the old pre k teacher was so excited to tell me DS was playing more with the other kids at center time, curious I asked DS if he had read all the books in the class,he had, so was now going to play with the kids. They had 2 sets of center time in a 6 hour day and he was SOOOO dome with it but in 3 hours they only had 1 block of it plus outdoor time.
I think you already know what to look for - adults that see the individual child, lots of toys, at varying levels, including beyond the level of the room, I would recommend a place that considers itself academic or developmentally play based, who use the language to sell what they do, mainly because they will have directed activities with goals in mind - lacing for motor skills - etc. And that will bring in the new data oddly for my DS even when he already knew what they were teaching.
So for me, the physical room is important, the flow of the day, the approach of the school, but ultimately it was the teachers and how they handle the quirky kid who prefers their company because he is totally uninterested in his peers
Wow, I keep writing these novels, not sure I provided anything useful, but I hope so

DeHe