I too think it would be best if Yoda remained a mystery and wasn’t a head of the council etc.
That he didn’t chose swampy forest because of an exile, but as a place to test himself, or a seclusion while he explores the Force, entire place for younger jedies should be sort of a pilgrimage where he decides are they worthy or not to be trained by him.
Didn’t he say he had a council on who was to be trained in ESB though?
“My own counsel will I keep, on who is to be trained!”
Counsel =/= Council