I wonder if all the services are accessible to everyone equally in all the countries that some people suggest are much better than ours in providing health care. One of the factors that led to the UK being ranked first in the research in 2014 was equality of access to health care.
It is, I realise, an extreme example but In the US those with vast amounts of money are happy with the system because they have access to more sophisticated treatments and procedures than we are likely to have in the UK. But if anyone has seen the Michael Moore documentary "Sicko", it showed examples of how when people became unemployed and lost their employment-related health insurance they could not access anything more than very basic health care. Expensive drug treatment was withheld in the case of chronic conditions and illnesses until they became acute. One woman related how her husband was denied expensive drug treatment for an ongoing condition and became so ill and debilitated that he eventually died, leaving her with young children.
As for charging for GP appointments, this would be a very cumbersome and costly system to administer. Also, some people on low incomes would be reluctant to go to the doctor and might possibly wait until it was too late for diagnosis and treatment to make any difference. And, in my experience, any charges that are introduced at one rate, rarely stay at that rate but instead increase year on year. This has happened with the controlled parking charges that my daughter pays, which have increased from a fairly small yearly payment to a quite substantial amount now.
The chain of private walk-in doctors' surgeries near mainline London and other city stations (not the non-chain, Harley Street-type ones that are much more expensive) are charging upwards of £55 per 15-minute appointment. I had a one-off appointment with a BUPA doctor near King's Cross because my foot was painful when I walked. She asked me to walk across the room, said I had flat feet and needed specialist, custom-made insteps - recommending where I could obtain them. That was four years ago. I paid £40 for what turned out to be a 5 minute consultation. (Incidentally, I didn't get the £350 (!) insteps she recommended and (touch wood) the problem rectified itself in due course. Once we accept that certain areas of health care can, as a matter of course, be charged for, this is, I think, a very dangerous road to go down.
Britain is rich and it is a country which had the vision to recognise that a civilised and successful country must look after all of its people. The idea of some sort of two-tier system that will favour the better off and discourage the less well off from seeking the treatment they need goes against everything that the NHS was created for.