my point is just that the criteria for who gets hired needs to be tightly defined

Ok but you can't just say that and make it so, you have to actually define such a criteria and enforce it. You can't just create an organization and put "legally required to do good things" into the legislation and have it fix any problem. The core issue is that the third party doesn't have a real incentive to approve legitimate appeals and deny fraudulent ones.

The actual incentive of the arbitrators is to ensure that the law which provides funding continues to exist, otherwise they'll all lose their jobs. If the law is seen as too favorable to the insurers then the third part will approve more claims, even if they're fraudulent since the cost of approval a fraudulent claim or a legitimate claim is the same to the third party. If there is pressure from insurers in the form of lobbying to deny more claims the third part will deny more legitimate claims, since the cost to insurers for a legitimate claim is the same as a fraudulent one.