I agree @Getoff I have no idea why you would charge for hospital stays instead of GP appointments.
The way it works in France is that the government subsidises 70% of the cost for GP appointments for everyone. This leaves about £20 left to pay for.
If you are on benefits or low income, the government pays that £20 for you.
Then what most people do is buy a mutuelle, a top-up health insurance policy. When we lived in France this was about £40/month for our family but it covered literally almost everything -- GP visits, prescriptions, hospital stays (even got you private rooms), dental, eyeglasses, physio, everything.
I know a lot of people in the UK would be viscerally unhappy with the idea of paying a monthly fee, but when you consider that it would cover things you currently pay for (dental, vision, prescriptions, various therapies) and hopefully improve overall quality, maybe it would be more acceptable?