It isn't about being cruel or heartless.
It's difficult to know what has happened here, but the statement from the LEA seems to imply that this was an infant class size case. If that is so, the appeal can be won only if the LEA made some sort of mistake or if the panel views the decision as so unreasonable that no rational LEA would have made the same decision.
We do not know whether the school/LEA has an admissions category for social/medical need and whether the family submitted evidence to support an application under that heading. If they did, then the LEA could have been held to be in error in ignoring it. If the younger sibling's health issues were only mentioned at the appeal stage, though, the LEA can't be held to have been at fault in not acting on them.
As tiggytape says, the law is very clear here and lobbying by MPs or community groups won't make any difference - the admissions code and appeal code have to be followed. I am sure the appeal panel will have felt desperately sorry for the family, but sometimes there are situations where appeal panels would like to help but simply don't have any way to do so.
If the family feel the appeal panel misdirected itself, they could refer the matter to the Local Government Ombudsman. Presumably, if there is a social/medical category for admissions, their daughter is now high on the waiting list.
I wish them well.