If this is how appeals work at your school, someone needs to retrain the appeal panel. It is entirely wrong. If an appeal panel refuses to hear arguments about PAN unless a mistake has been made, the parents can refer the case to the LGO or ESFA (depending on the type of school) and will automatically get a fresh hearing with a different appeal panel for such an egregious breach of the Appeals Code.
I suspect this isn't how appeals actually work and you are simply demonstrating that most head teachers don't fully understand the process.
Appeals are indeed in two stages, but they are not as you describe. In stage 1 the school argues that the correct process has been followed and the school cannot possibly accommodate any more pupils. This is where arguments about whether the school can go over PAN without prejudice are heard. Questions as to whether a mistake has been made may crop up at this stage, but they may crop up in stage 2. In some cases, stage 1 is heard as a grouped appeal, with all appellants present. In that situation, arguments about mistakes will almost always be deferred to stage 2 unless the mistake affects multiple appellants.
Stage 2 is where the parents put forward their arguments that a mistake has been made and/or their child will suffer prejudice if not admitted.
After stage 2 of the hearing, the panel follows a two-stage process to make its decision. First they decide if a mistake has been made. If they find that a mistake has been made, the child is admitted unless the mistake affects multiple children and the school couldn't cope if they were all admitted. In that situation, the appeal panel compare cases to decide which children to admit. If no mistake has been made, the panel decide if the school will be prejudiced by the admission of additional pupils. This is still part of stage one of the decision-making process. If they find there will be no prejudice to the school, the appeals all succeed. However, that is rare. Normally, the appeal panel find that there will be some prejudice, so they move on to the second stage of the decision-making process.
In the second stage, the panel decide if the prejudice to the child from not being admitted outweighs the prejudice to the school.
There are no circumstances in which the waiting list enters into the decision. If the appeal panel decide that the school can admit five pupils without prejudice, they will admit the five appellants with the strongest cases even if there are 100 people ahead of them on the waiting list.
The process is identical for primary and secondary schools unless infant class size rules apply (which, of course, can only happen at primary schools). The hearing still follows the same process, but the decision-making process is different. In this case, stage one of the decision-making process is about whether the admission authority is correct that infant class size rules apply and, if so, whether a mistake has been made that cost an appellant a place. If it is an infant class size case and a mistake has been made, they move on to stage two which involves comparing cases to decide which appeals to uphold.
What you have described is similar to the decision-making process for infant class size appeals, but it is not what happens in the hearing for any appeal. There are no circumstances whatever in which the appeal panel can refuse to hear the parents' arguments about whether the school can admit more children. Any refusal to hear part of the parents' case or to allow them to put all the questions they want automatically means the appeal is unfair and the parents are entitled to a fresh hearing if they lose.