If they think it's a big enough problem I think they could screen out most of the cheaters without making it so cumbersome as to deter too many of the valid cases. Following your suggestion, require an application in advance, where you have to list the contact info for a doctor that can verify the condition. Very few cheaters will have the nerve to apply if they think there's a chance they might show up for their card at Guest Relations only to be told face to face that their info didn't check out as truthful. It wouldn't take much actual spot checking to make the system credible. Maybe check out 1 in every 50 or so applications. Keep a database so anyone who is verified once is permanently verified.
This would cost Disney something, so it all depends on just how serious they think the cheating problem is. They shouldn't assume there are a lot of cheaters just because they issue a lot of cards. Disney has a reputation for being disability-friendly so maybe we just go more often than disability-free families.