What I was wondering about was how much money is needed for the sheer administration of processing claims and payments in the US system?

In the UK in the NHS, there is none of the layers of beurocracy and paperwork related to call-centers and payment handling and claims inspectors (as far as I know anyway) that I assume is commonplace in the US hospitals and insurance companies. You go in, see the doctor, then leave. Usually the only "paperwork" required is booking another appointment if you need to come back - no forms, no invoices, no payment etc.

I guess it is the same argument made about universal income benefits - if we just gave everyone (rich or poor) in the country a set amount of money each year we'd save billions in not having to means-test every claimant individually and actually save money by giving it away to everyone.