Having worked at a few GP practices, there does seem to be variation on how things are run. A lot is based on how to manage the "demand". IIRC a study showed that if a GP saw every patient that wanted to be seen there and then by them, they'd have to work 21-hour days. Clearly impossible, so there has to be some form of rationing. The larger the practice, the harder it is to ration - hence you have things like telephone triage.
There does appear to be a loss of basic knowledge on how to look after simple conditions, and I think this may be down to the loss of the traditional family unit whereby the elders in the family would know how to deal with a simple headache or sore throat, and when a doctor really was required. On some days, I reckon 25% of what I see is either self-limiting or easily manageable by the patient themselves.
As for charging, I'm against this on the basis of: 1. it goes against the principle of the NHS, 2. it would dissuade those who really needed to be seen from attending, and 3. it alters the dynamics of the consultation. Someone paying x amount for a consultation would want to come away with something whether it be a referral or a prescription, even if it were for a simple viral illness. People are also more likely to demand more "bang for their buck" and want to deal with several issues in the same 10-minute slot.
Interestingly, a private GP can charge around £70-80 for a single 15-min consultation. In the NHS, a GP is paid around £65-70 for each patient per year, irrespective of the number of consultations.