I can see from this website that complaints about Academy appeals should go to the EFA. Is this still the case where the Academy has outsourced their appeals process to the Local Education Authority?
Yes, it's the EFA.
Also for a non qualification appeal, should the letter received from the Appeal Clerk make specific reference to the actual evidence submitted
and why it was not sufficient or is it acceptable for them to merely state that what was submitted was "insufficient to prove academic ability and the appeal is therefore unsuccessful"?
It's possible that the clerk's notes might give more information, if you can get hold of a copy.http://www.elevenplusexams.co.uk/appeals/ombudsman#d4
If the clerk's notes show that your key points were
properly considered, then their omission from the decision letter would be a technical breach of 2.25 of the Code (as opposed to what might otherwise be a serious injustice).