Applying under the exceptional medical or social needs admission rule (EMSAR) | Buckinghamshire Council https://share.google/jwS9isDpUh9vTDuze
Looking at this, the scope of the panel is to judge whether an application to a school which fall under its remit and actually has 'EMS needs' as an oversubscription criterion (and is the applicant's first preference school, plus one lower ranked upper school where the first preference is a grammar) should or should not be ranked under this criterion. It's not, 'should a school which doesn't have this as an oversubscription criterion be told to give this child priority'.
I don't know whether there panel is told that the school(s) is / are the highest preference(s), but presumably not, if the school the OP submitted evidence for is their third preference grammar, the only one with such an oversubscription criterion, and the response from the panel was 'Evidence does not meet threshold', rather than, 'We're not looking at this, because the named school doesn't have this criterion', or 'We're not looking at this, because although the named school does have this criterion, it was not the applicant's highest ranked grammar school on their CAF?
@busySunflower you say that you put your reasons in the free text box, but for the school which does include the EMS criterion in its admissions policy, was there not a question on the CAF, once you selected the school, along the lines of, 'Are you applying under criterion X? If so, you need to do this...?
If you are not applying from within Buckinghamshire County Council, then I can see that there might be an issue, but there should have been something in your home LA's 'essential information for parents' regarding checking each school's policy for any additional paperwork to be submitted
Admissions law allows parents to state any reason they like as to why they want a particular school, but equally, it restricts schools to only using its published oversubscription criteria to rank applications.