where, exactly, it says that training, other than for the issue of a licence or rating, has to be conducted by a Flight Instructor
It doesn't say it anywhere however; it is "flying training" that is exempted from the requirement to hold an AOC, so if its not Flying Training as defined in the ANO, then it can only be a Private Flight where the pilot must pay his share, or its Public Transport and requires an AOC. The choice is simple!