We are actually setting up an insurer in the UK which is doing exactly that by changing the business model to taking fees on settled insurance claims instead of betting on an underwriting/investment profit.
Have a look.
insureathing.com
Have a look.
insureathing.com
"All I've heard so far are physicians who don't accept insurance but instead have a straightforward "menu" for common items, which is interesting but not what I think most people want." - I think people want this but they are scared of going off of insurance in the event they need to see someone who doesn't offer this (chance occurance, expensive disease).