Well, coming from a country where doctors' appointments are subsidised, but there is a charge, I do think that perhaps there is a case to be made for charging for appointments, to weed out those that are not necessarily requiring a doctor's time.
And then appointments can be made free of charge for children, up to X age (preferably school leaving age), and an eligibility card because of income, physical or mental circumstances, etc, can significantly reduce the charge, or make it free again.
But I find it really odd, and a tad uncomfortable, that we don't pay for GP's appointments. We have an income that means that we could, (and I feel, should). Of course, that means we're paying a truckload in tax, too, so I suppose we could say, well, we pays our taxes, we're due something back, aren't we? I'd rather not though. We do try to access healthcare as little as possible. When we've had to, gosh, the care and the compassion have been fantastic.