Apart from your idea of signing up to a specific hospital this is how systems like the NHS work. They are funded from a specific tax in the UK and general taxa

Apart from your idea of signing up to a specific hospital this is how systems like the NHS work. They are funded from a specific tax in the UK and general taxation in some other countries. They either don't bill anyone for the care they provide and have no bureaucratic machinery for doing so, like the UK NHS or a here in Norway we pay a sort of co-pay for each doctor or hospital visit but it is capped at quite low levels, a couple of hundred dollars a year. And it is not related to the fine details of the treatment you have received so again there is very little bureaucracy.