has anyone actually won on the ground that the decision not to give the child a place was so perverse
To win on that basis the decision has to be Wednesbury unreasonable, i.e. "So outrageous in its defiance of logic or accepted moral standards that no sensible person who had applied his mind to the question to be decided could have arrived at it". Missing out because there were 3 sets of twins this year doesn't come remotely close to meeting that bar.
it is hugely unlikely that there will be over 30 children next year, the village just isn't very big. Can we use that as an argument?
No. If PAN is 30 this will be an infant class size case. That means you can only win if the admission arrangements did not comply with the Admissions Code, the arrangements were not implemented correctly or the decision to refuse admission was unreasonable. The panel cannot admit on the basis of speculation about what may or may not happen next year.
Our reasons are that it's the only school we can walk to
I'm afraid that won't help you at appeal. The panel cannot consider transport or childcare issues. Your arguments around social exclusion are also unlikely to persuade the appeal panel.
The school we've been offered is 3 miles away
If it is over 2 miles by the shortest safe walking route your son will be entitled to free transport to and from school.
It seems they have measure from the main school gate to a point at each child's address
That will be defined in the admission criteria. As this is an academy they may say that they use the distance as determined by the LA's computerised system. That will almost certainly use the address point defined by Royal Mail which is usually in the middle of the property. You will not win an argument that they have measured from the wrong place on your house. You would have a better chance if it is unclear which is the main school gate or if they have measured from the wrong house.
As far as I can work out, no-one who has been offered a place has been asked to prove that the address they gave is actually where they live
Most LAs ask for some proof at application time. Whether they do or not you are unlikely to win an appeal on this basis. If you have evidence of a fraudulent application you should notify the LA.
Does anyone know what kind of reasons do qualify as perversely unreasonable?
Something like the child being placed in danger by going to this school, e.g. the parent is a police officer or social worker and the school is used by families with whom the parent is professionally involved.
There shouldn't be any who live further away who got a place on the sibling criteria as that is ranked below proximity to school here
Do you mean the school has a defined catchment area and people living within catchment get priority over out of catchment siblings? If that isn't what you mean I struggle to see how that would work.