I also think a small copay for visits isn't a bad idea. Nominal cost - £5 max. But something to make people think before booking a visit. But then part of me hates this idea as I wouldn't want £5 to prevent someone from booking a necessary appointment.
You will pay a lot more than £5 a visit in other countries.
Try 60E a GP visit each time you or your children age 5 plus went to see a GP. 100E for each A&E visit. No cheap prescriptions, but capped at about 140E a month for each person. Plus a monthly fee to use the health service and taxes to pay. That's Ireland, but I doubt there are many countries where there is no fee to see the GP, or a medical to pass to get a visa. Why do you think the UK's NHS is so popular with foreign nationals and with Brits returning years later for retirement?
Some wealthy EEA countries require their elderly to have private insurance too if they want some treatments/operations, as well as pay the monthly insurance to the national scheme.
Have you missed what has just been going on in Germany and their healthcare system? Everyone pays monthly into their health system and pays taxes, but the higher skilled get to see a doctor quicker as their employers give them private insurance too and doctors can bill these for more money. The socialist party wanted to change this but only got about 20% of the vote in the election and scored their lowest vote in the last 40 years. While the talks were on with Merkel to form a coaliton to try to change this healthcare system, the polls showed that the socialist party has now dropped below the far right party AfD.
We see this in other western countries too with some lower skilled immigrants who won't be paying a lot of taxes to that country. They all have to pass a medical to get a visa, but they get less healthcare given bill free than a citizen of that country and that family need to buy private insurance too. They all pay for each visit to a doctor, even for children, as citizens have to do that too. Plus these immigrants can be charged a fee for each term, for each child they send to a state funded school.