There is, of course, an easy solution.
The introduction of co-payments is the only intervention ever to have shown to reduce use of primary care services. (1) This could be implemented in a proportional way, lower co-pays for lower income patients. Lower co-pays or rebates for preventive interventions e.g. diabetes and hypertension control.
Payments and records could be linked to the inland revenue, and administered through an NHS card (biometrics) that also establishes eligibility for NHS care.
Doctors could still be paid on capitation or salary, removing supplier induced demand, and patient can be charged according to use, incentivising efficient use of resources.
A modern day example the use of carrier bags after the 5 pence fee.
I think we also need some good data on the reasons why patients are requesting an appointment to see their GP. This could help us better understand what might be necessary when it comes to supporting prevention. If, for example, the data shows that a significant percentage of people are wishing to see their GP because of a 'lifestyle' related condition, this would give even more reason for developing policies that encourage prevention. This could then help to free up the system, put less press on the GPs, and also people with medical conditions such as arthritis or Parkinson's are able to get better access their GP. But also, those people with symptoms suggestive of a neurological conditions are able to get quicker access to specialist services, as they need time with the GP for an assessment and if necessary a referral.