Whether medlink was informed accurately about the symptoms is not going to be known since the person who informed them is not ( afaik ) on this forum

The attendant may have given an accurate description of the symptoms on arrival but what was reported to passenger misrepresented what the attendant said

What I do see though, is that if medlink advised that the passenger was too ill to fly, that it would be very strange for a flight captain to override this advice and so do not see that the captain did anything wrong at all in refusing travel

If the passenger's parent was a medical doctor, she may have been able to diagnose the child as being ok to travel, but being a head teacher of a school does not make her qualified to provide an "ok to fly" certificate