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?
Should Vader and The Emperor even know who Yoda is?
Nope. I always assumed they had no clue, otherwise they would have attacked Dagobah and tried to eliminate him. Vader mentions Obi-Wan’s name repeatedly, but never mentions Yoda in the OT.
It’s heavily implied that Yoda has come into contact with both of them before. In ESB he talks about how strong Luke’s father was, and in ROTJ he tells Luke to not underestimate the Emperor. He went into hiding on some random remote planet, it’s much less far fetched that they never went after him as opposed to Vader going after Luke who’s “hiding” with his family.