There are schools that can serve in that manner, but it’s a tiny minority of them. No (or virtually no) public schools with 125+ students per grade will have that tight community structure.
A private school with 20 per class and 60 per grade has a fair shot at it. Maybe a small public district could as well, but I’ve never seen it happen there.
Sports teams with strong non-athletic aspects to their program are another possible source of values transmission.
I agree that it’s not the sign on the building that matters, but the content and consistency of what happens inside it.
I daresay the label of the community is irrelevant, what matters is some other aspect effective ones share - and of course, the child in question (: