Back to the question about being asked for other schools applied for - like @PatriciaHolm , the paperwork sent to panels on which I sit (organised under the umbrella of Democratic Services within the LA) and to the parent, includes a copy of the CAF or mid-year application form in its document bundle. A major point of this is to assist us in considering whether there may have been an error in the allocation process. e.g. parent has named a sibling in year 9 at their first preference school, so child should have been ranked under Criterion 3 for that particular school, and, also according to the school paperwork, offers were made down to Criterion 5, so a place should have been offered. But in error this wasn't taken into consideration and the school is ranking the child under Criterion 6. An error in the process like that which led to that specific pupil being denied a place would mean the appeal being upheld. Hopefully, a 'wholesale' error -- would be picked up and dealt with before the appeal hearings stage.
Personally, although at each hearing, it is only Why must your DC have a place at this particular school? that is the question, knowing that this school had only been ranked third / fourth / last on the CAF wouldn't put out of my mind that it is the parents' right to appeal for any school which has refused their application.
(However, I will admit that, Seriously? might well be my first thought at many reasons - or 'reasons' - given, at least on first sight of the appeal form. Supporting democratic rights and having views on doing things just because one can are not mutially exclusive...).
@CoffeeAndCommas have you now applied for the other local school? If it would have been 'okay' and you are within its defined catchment area, it would be an idea to apply (and being turned down, if you are, will give you the right of appeal for this one, as well; you can always withdraw from other appeals before the hearing, if a place becomes avaliable)