This is a complaint to the DofE and nearly everyone said in the first place she had no grounds
Whilst a number of people said she had no grounds for an appeal, those with expertise (PanelChair, PatriciaHolm and Lougle) were all of the view that she potentially has a case. I agree with them. If the appeal school has better provision for young carers than the allocated school, OP clearly has a case.
And the information OP has posted shows that she may well have grounds for a complaint to the DfE. For a start, the panel appear to have decided to reject the appeal on the grounds that they were not convinced that the appeal school was the only school that can meet her child's needs. If that was their approach, it is clearly wrong. The only question is whether the appeal school would meet her child's needs better than the allocated school. The fact that some other schools may be equally able to meet her child's needs is irrelevant. The other schools may not have places available.
There is also a suspicion in the decision letter that the appeal panel may have effectively outsourced their decision making to the panel that decided which children met the school's medical and social needs criteria. If they did, that is also clearly wrong. They are required to make their own decision, not rely on someone else's.
Also, the panel excluded some of OP's evidence on spurious grounds, arguing that she should not have had information which the school would have been required to provide if requested and which is, in any case, publicly available. Excluding this information may have damaged OP's case.
I don't know if her reference to DfE will succeed as I haven't seen all the information, but there is certainly enough in OP's posts to suggest she may have a case for a fresh hearing with a different panel.